On computer algebra-aided stability analysis of dierence schemes generated by means of Gr obner bases

Transcription

1 On computer algebra-aided stability analysis of dierence schemes generated by means of Gr obner bases Vladimir Gerdt 1 Yuri Blinkov 2 1 Laboratory of Information Technologies Joint Institute for Nuclear Research 2 Department of Mathematics and Mechanics Saratov State University 23 Feb 2007

2 Outline Introduction Finite Dierence Approach Stability of Dierence Schemes Dierence schemes for hyperbolic equations Dierence Cauchy problem Notion of approximation for the initial problem First dierential approximation of dierence scheme Example: Lax Scheme Algorithmic Approach to Generation of Dierence Schemes Algorithm for Construction of Dierential Approximation Algorithm for Hyperbolic Form Algorithm for Hyperbolic Form Implementation in Maple Two-Step Lax-Wendro Schemes Conclusions Bibliography

3 Outline Introduction Finite Dierence Approach Stability of Dierence Schemes Dierence schemes for hyperbolic equations Dierence Cauchy problem Notion of approximation for the initial problem First dierential approximation of dierence scheme Example: Lax Scheme Algorithmic Approach to Generation of Dierence Schemes Algorithm for Construction of Dierential Approximation Algorithm for Hyperbolic Form Algorithm for Hyperbolic Form Implementation in Maple Two-Step Lax-Wendro Schemes Conclusions Bibliography

4 Outline Introduction Finite Dierence Approach Stability of Dierence Schemes Dierence schemes for hyperbolic equations Dierence Cauchy problem Notion of approximation for the initial problem First dierential approximation of dierence scheme Example: Lax Scheme Algorithmic Approach to Generation of Dierence Schemes Algorithm for Construction of Dierential Approximation Algorithm for Hyperbolic Form Algorithm for Hyperbolic Form Implementation in Maple Two-Step Lax-Wendro Schemes Conclusions Bibliography

5 Outline Introduction Finite Dierence Approach Stability of Dierence Schemes Dierence schemes for hyperbolic equations Dierence Cauchy problem Notion of approximation for the initial problem First dierential approximation of dierence scheme Example: Lax Scheme Algorithmic Approach to Generation of Dierence Schemes Algorithm for Construction of Dierential Approximation Algorithm for Hyperbolic Form Algorithm for Hyperbolic Form Implementation in Maple Two-Step Lax-Wendro Schemes Conclusions Bibliography

6 Outline Introduction Finite Dierence Approach Stability of Dierence Schemes Dierence schemes for hyperbolic equations Dierence Cauchy problem Notion of approximation for the initial problem First dierential approximation of dierence scheme Example: Lax Scheme Algorithmic Approach to Generation of Dierence Schemes Algorithm for Construction of Dierential Approximation Algorithm for Hyperbolic Form Algorithm for Hyperbolic Form Implementation in Maple Two-Step Lax-Wendro Schemes Conclusions Bibliography

7 Outline Introduction Finite Dierence Approach Stability of Dierence Schemes Dierence schemes for hyperbolic equations Dierence Cauchy problem Notion of approximation for the initial problem First dierential approximation of dierence scheme Example: Lax Scheme Algorithmic Approach to Generation of Dierence Schemes Algorithm for Construction of Dierential Approximation Algorithm for Hyperbolic Form Algorithm for Hyperbolic Form Implementation in Maple Two-Step Lax-Wendro Schemes Conclusions Bibliography

8 Outline Introduction Finite Dierence Approach Stability of Dierence Schemes Dierence schemes for hyperbolic equations Dierence Cauchy problem Notion of approximation for the initial problem First dierential approximation of dierence scheme Example: Lax Scheme Algorithmic Approach to Generation of Dierence Schemes Algorithm for Construction of Dierential Approximation Algorithm for Hyperbolic Form Algorithm for Hyperbolic Form Implementation in Maple Two-Step Lax-Wendro Schemes Conclusions Bibliography

9 Finite Dierence Approach The nite dierence approach is the most popular discretization technique for numerical solving of ordinary or PDEs. In this approach derivatives are approximated by nite dierences and the resulting algebraic system dierence scheme is solved numerically. Recently (G.,B.,Mozzhilkin'06) we developed an algorithmic method to generation of nite dierence schemes for linear PDEs with two independent variables. The method is based on dierence elimination provided by construction of Gr obner bases for an appropriate elimination ranking. Sometimes Gr obner bases can be computed even for nonlinear dierence systems obtained by discretization of PDEs and related integral equations. In this case nonlinear dierence schemes can also be generated by our method.

10 Finite Dierence Approach The nite dierence approach is the most popular discretization technique for numerical solving of ordinary or PDEs. In this approach derivatives are approximated by nite dierences and the resulting algebraic system dierence scheme is solved numerically. Recently (G.,B.,Mozzhilkin'06) we developed an algorithmic method to generation of nite dierence schemes for linear PDEs with two independent variables. The method is based on dierence elimination provided by construction of Gr obner bases for an appropriate elimination ranking. Sometimes Gr obner bases can be computed even for nonlinear dierence systems obtained by discretization of PDEs and related integral equations. In this case nonlinear dierence schemes can also be generated by our method.

11 Finite Dierence Approach The nite dierence approach is the most popular discretization technique for numerical solving of ordinary or PDEs. In this approach derivatives are approximated by nite dierences and the resulting algebraic system dierence scheme is solved numerically. Recently (G.,B.,Mozzhilkin'06) we developed an algorithmic method to generation of nite dierence schemes for linear PDEs with two independent variables. The method is based on dierence elimination provided by construction of Gr obner bases for an appropriate elimination ranking. Sometimes Gr obner bases can be computed even for nonlinear dierence systems obtained by discretization of PDEs and related integral equations. In this case nonlinear dierence schemes can also be generated by our method.

12 Stability of Dierence Schemes A dierence scheme, to be of practical interest, must be stable. The stability study of dierence schemes exploits symbolic mathematical operations. Thus it can be analyzed with help of computer algebra methods and software (Ganzha,Vorozhtsov'96). To analyze stability one can use dierential approximation that is often called the modied equation(s) of the dierence scheme. There are whole classes of dierent schemes for which their stability properties can be obtained with the aid of the dierential approximation (Strikwerda'04). For all that, in many cases, the computation can be done by means of modern computer algebra software. In this talk we shall demonstrate how Maple can be used for this purpose and present a Maple program for computation of dierential approximations for dierence schemes. In the aggregate with the Maple package for construction of Gr obner bases for linear dierence systems (G.,Robertz'06) the program allows one to generate schemes possessing stability properties.

13 Stability of Dierence Schemes A dierence scheme, to be of practical interest, must be stable. The stability study of dierence schemes exploits symbolic mathematical operations. Thus it can be analyzed with help of computer algebra methods and software (Ganzha,Vorozhtsov'96). To analyze stability one can use dierential approximation that is often called the modied equation(s) of the dierence scheme. There are whole classes of dierent schemes for which their stability properties can be obtained with the aid of the dierential approximation (Strikwerda'04). For all that, in many cases, the computation can be done by means of modern computer algebra software. In this talk we shall demonstrate how Maple can be used for this purpose and present a Maple program for computation of dierential approximations for dierence schemes. In the aggregate with the Maple package for construction of Gr obner bases for linear dierence systems (G.,Robertz'06) the program allows one to generate schemes possessing stability properties.

14 Stability of Dierence Schemes A dierence scheme, to be of practical interest, must be stable. The stability study of dierence schemes exploits symbolic mathematical operations. Thus it can be analyzed with help of computer algebra methods and software (Ganzha,Vorozhtsov'96). To analyze stability one can use dierential approximation that is often called the modied equation(s) of the dierence scheme. There are whole classes of dierent schemes for which their stability properties can be obtained with the aid of the dierential approximation (Strikwerda'04). For all that, in many cases, the computation can be done by means of modern computer algebra software. In this talk we shall demonstrate how Maple can be used for this purpose and present a Maple program for computation of dierential approximations for dierence schemes. In the aggregate with the Maple package for construction of Gr obner bases for linear dierence systems (G.,Robertz'06) the program allows one to generate schemes possessing stability properties.

15 Stability of Dierence Schemes A dierence scheme, to be of practical interest, must be stable. The stability study of dierence schemes exploits symbolic mathematical operations. Thus it can be analyzed with help of computer algebra methods and software (Ganzha,Vorozhtsov'96). To analyze stability one can use dierential approximation that is often called the modied equation(s) of the dierence scheme. There are whole classes of dierent schemes for which their stability properties can be obtained with the aid of the dierential approximation (Strikwerda'04). For all that, in many cases, the computation can be done by means of modern computer algebra software. In this talk we shall demonstrate how Maple can be used for this purpose and present a Maple program for computation of dierential approximations for dierence schemes. In the aggregate with the Maple package for construction of Gr obner bases for linear dierence systems (G.,Robertz'06) the program allows one to generate schemes possessing stability properties.

16 Dierence Cauchy problem Consider the following Cauchy problem u = Au, < 0 <, t > 0 (1) t u(x, 0) = u 0 (x), < 0 <, (2) where x is the spatial variable, t is the temporal variable, A is a linear dierential operator, u 0 (x) is a given function. We approximate the Cauchy problem (1), (2) by the following dierence Cauchy problem: u n+1 j u n j = Λ 1 u n+1 j + Λ 2 u n j, j = 0, ±1, ±2,... ; n = 0, 1, 2,... (3) u 0 j = u 0 (x j ) (4)

17 Dierence Cauchy problem Consider the following Cauchy problem u = Au, < 0 <, t > 0 (1) t u(x, 0) = u 0 (x), < 0 <, (2) where x is the spatial variable, t is the temporal variable, A is a linear dierential operator, u 0 (x) is a given function. We approximate the Cauchy problem (1), (2) by the following dierence Cauchy problem: u n+1 j u n j = Λ 1 u n+1 j + Λ 2 u n j, j = 0, ±1, ±2,... ; n = 0, 1, 2,... (3) u 0 j = u 0 (x j ) (4)

18 Notion of approximation Let L be the operator of the initial equation (1), i.e. Lu = u Au, (5) t and let L h be a dierence operator dened in accordance to (5) as L h u = u(x, t + ) u(x, t) Λ 1 u(x, t + ) Λ 2 u(x, t), (6) Let u(x, t) be a solution of the Cauchy problem (1), (2) smooth enough. If Lu L h u C 1 h k 1 + C 2 k 2, (7) where k 1 > 0, k 2 > 0 and constants C 1 è C 2 do not depend on and h, then (by denition) dierence scheme (3) approximates equation (1) and has order of approximation k 1 w.r.t. h and order k 2 w.r.t..

19 Notion of approximation Let L be the operator of the initial equation (1), i.e. Lu = u Au, (5) t and let L h be a dierence operator dened in accordance to (5) as L h u = u(x, t + ) u(x, t) Λ 1 u(x, t + ) Λ 2 u(x, t), (6) Let u(x, t) be a solution of the Cauchy problem (1), (2) smooth enough. If Lu L h u C 1 h k 1 + C 2 k 2, (7) where k 1 > 0, k 2 > 0 and constants C 1 è C 2 do not depend on and h, then (by denition) dierence scheme (3) approximates equation (1) and has order of approximation k 1 w.r.t. h and order k 2 w.r.t..

20 First dierential approximation of dierence scheme The rst dierential approximation (FDA) of dierence schemes (3) is the partial dierential equation which is obtained from (3) by substitution for the grid function their Taylor expansions and by keeping the main term (Shokin,Yanenko'85) One distinguishes hyperbolic and parabolic forms of FDA (Ganzha, Vorozhtsov'96). To obtain parabolic form of FDA one uses dierential consequences of the initial PDE(s) u t = Au. as a result of dierentiation of the both sides of PDE(s) w.r.t. the independent variables.

21 First dierential approximation of dierence scheme The rst dierential approximation (FDA) of dierence schemes (3) is the partial dierential equation which is obtained from (3) by substitution for the grid function their Taylor expansions and by keeping the main term (Shokin,Yanenko'85) One distinguishes hyperbolic and parabolic forms of FDA (Ganzha, Vorozhtsov'96). To obtain parabolic form of FDA one uses dierential consequences of the initial PDE(s) u t = Au. as a result of dierentiation of the both sides of PDE(s) w.r.t. the independent variables.

22 Error of Dierence Scheme Discretization of a PDE implies that a dierence solution does not satisfy PDE. The deviation of dierence solution from the exact one is called an error of dierence scheme Study and classication of errors is based on representation of the solution by a trigonometric Fourier series and detectiing the variation in amplitude and phase of each harmonic in one step in time and, respectively, the variation of the exact solution (of PDE) on the same time interval. If the harmonic amplitude decreases faster then that for the exact solution, then this eect is called the amplitude error of the scheme caused by an extra diusion inherent to the scheme numerical viscosity. The phase variation of the dierence solution distinct from that for the exact solution is called the phase error caused by distinction in the phase velocities of the harmonic propagation numerical dispersion.

23 Error of Dierence Scheme Discretization of a PDE implies that a dierence solution does not satisfy PDE. The deviation of dierence solution from the exact one is called an error of dierence scheme Study and classication of errors is based on representation of the solution by a trigonometric Fourier series and detectiing the variation in amplitude and phase of each harmonic in one step in time and, respectively, the variation of the exact solution (of PDE) on the same time interval. If the harmonic amplitude decreases faster then that for the exact solution, then this eect is called the amplitude error of the scheme caused by an extra diusion inherent to the scheme numerical viscosity. The phase variation of the dierence solution distinct from that for the exact solution is called the phase error caused by distinction in the phase velocities of the harmonic propagation numerical dispersion.

24 Lax-type Scheme for Burgers' equation u t + f x = ν u xx, ν = const (8) 2u n+1 j+2 (un j+3 + un j+1 ) + fn j+3 fn j+1 = ν un j+4 2un j+2 + un j 2 2h 4h 2 (9) Dierential approximation in point (n, j + 2) { }} { u t + f x νu xx 1 2 u xx h u tt + ( 1 6 f xxx 1 3 νu xxxx)h u xxxx h u ttt 2 + ( f xxxxx 2 45 νu xxxxxx)h = u xxxxxx h6 + (10) From (10) it follows that scheme (9) does not approximate equation óðàâíåíèå (8) at O(h 2 /) 1. It is an example of conditionally convergent scheme.

25 Lax-type Scheme for Burgers' equation u t + f x = ν u xx, ν = const (8) 2u n+1 j+2 (un j+3 + un j+1 ) + fn j+3 fn j+1 = ν un j+4 2un j+2 + un j 2 2h 4h 2 (9) Dierential approximation in point (n, j + 2) { }} { u t + f x νu xx 1 2 u xx h u tt + ( 1 6 f xxx 1 3 νu xxxx)h u xxxx h u ttt 2 + ( f xxxxx 2 45 νu xxxxxx)h = u xxxxxx h6 + (10) From (10) it follows that scheme (9) does not approximate equation óðàâíåíèå (8) at O(h 2 /) 1. It is an example of conditionally convergent scheme.

26 Lax-type Scheme for Burgers' equation u t + f x = ν u xx, ν = const (8) 2u n+1 j+2 (un j+3 + un j+1 ) + fn j+3 fn j+1 = ν un j+4 2un j+2 + un j 2 2h 4h 2 (9) Dierential approximation in point (n, j + 2) { }} { u t + f x νu xx 1 2 u xx h u tt + ( 1 6 f xxx 1 3 νu xxxx)h u xxxx h u ttt 2 + ( f xxxxx 2 45 νu xxxxxx)h = u xxxxxx h6 + (10) From (10) it follows that scheme (9) does not approximate equation óðàâíåíèå (8) at O(h 2 /) 1. It is an example of conditionally convergent scheme.

27 Parabolic Form for FDA For more detailed analysis of scheme (9) one can construct a parabolic form of FDA for f = u 2 /2: { }} { u t + uu x (ν + h2 h2 + (2ν )u x ) 2 u2 u xx + +(u)u x (νu + h2 3 u)u xxx + ( ν2 2 + νh2 6 + h4 12 )u xxxx = 0. To be an approximation of the initial PDE it is necessary that expression marked by {}}{... to be ν whereas the remaining terms which do not occur in PDE to be vanish. For equation (8) and zero viscosity (ν = 0) the expression marked by {}}{... must be > 0. Otherwise the diusion coecient becomes negative. Thereby the boundary-value problem becomes incorrect. For some classes of PDE one can relate (equivalence theorem) the scheme stability with its dierential approximation. (11)

28 Parabolic Form for FDA For more detailed analysis of scheme (9) one can construct a parabolic form of FDA for f = u 2 /2: { }} { u t + uu x (ν + h2 h2 + (2ν )u x ) 2 u2 u xx + +(u)u x (νu + h2 3 u)u xxx + ( ν2 2 + νh2 6 + h4 12 )u xxxx = 0. To be an approximation of the initial PDE it is necessary that expression marked by {}}{... to be ν whereas the remaining terms which do not occur in PDE to be vanish. For equation (8) and zero viscosity (ν = 0) the expression marked by {}}{... must be > 0. Otherwise the diusion coecient becomes negative. Thereby the boundary-value problem becomes incorrect. For some classes of PDE one can relate (equivalence theorem) the scheme stability with its dierential approximation. (11)

29 Parabolic Form for FDA For more detailed analysis of scheme (9) one can construct a parabolic form of FDA for f = u 2 /2: { }} { u t + uu x (ν + h2 h2 + (2ν )u x ) 2 u2 u xx + +(u)u x (νu + h2 3 u)u xxx + ( ν2 2 + νh2 6 + h4 12 )u xxxx = 0. To be an approximation of the initial PDE it is necessary that expression marked by {}}{... to be ν whereas the remaining terms which do not occur in PDE to be vanish. For equation (8) and zero viscosity (ν = 0) the expression marked by {}}{... must be > 0. Otherwise the diusion coecient becomes negative. Thereby the boundary-value problem becomes incorrect. For some classes of PDE one can relate (equivalence theorem) the scheme stability with its dierential approximation. (11)

30 Parabolic Form for FDA For more detailed analysis of scheme (9) one can construct a parabolic form of FDA for f = u 2 /2: { }} { u t + uu x (ν + h2 h2 + (2ν )u x ) 2 u2 u xx + +(u)u x (νu + h2 3 u)u xxx + ( ν2 2 + νh2 6 + h4 12 )u xxxx = 0. To be an approximation of the initial PDE it is necessary that expression marked by {}}{... to be ν whereas the remaining terms which do not occur in PDE to be vanish. For equation (8) and zero viscosity (ν = 0) the expression marked by {}}{... must be > 0. Otherwise the diusion coecient becomes negative. Thereby the boundary-value problem becomes incorrect. For some classes of PDE one can relate (equivalence theorem) the scheme stability with its dierential approximation. (11)

31 Algorithmic Generation of Dierence Schemes In (G.,B.,Mozzhilkin'06) we suggested an algorithmic approach to generation of dierence schemes. By example consider equation (8). u t dt = u, u t = u n+1 j un j+2 + un j, 2 f x dx = f, = 2 h(f x ) n j+1 = fj+2 n fj n, (12) u x dx = u, 2 h(u x ) n j+1 = u n j+2 u n j, u xx dx = u x, 2 h(u xx ) n j+1 = (u x ) n j+2 (u x ) n j. Gr obner basis for u xx u t u x f x u f = scheme (9). If call such method of integration in time as Lax-type scheme, then one can use dierent numerical quadrature formulae for integration over x. For the midpoint or the trapezoidal rule for these integrals 8 dierent schemes are generated. Question: how close are properties of these 8 schemes? A partial answer can be obtained by using the dierential approximation technique.

32 Algorithmic Generation of Dierence Schemes In (G.,B.,Mozzhilkin'06) we suggested an algorithmic approach to generation of dierence schemes. By example consider equation (8). u t dt = u, u t = u n+1 j un j+2 + un j, 2 f x dx = f, = 2 h(f x ) n j+1 = fj+2 n fj n, (12) u x dx = u, 2 h(u x ) n j+1 = u n j+2 u n j, u xx dx = u x, 2 h(u xx ) n j+1 = (u x ) n j+2 (u x ) n j. Gr obner basis for u xx u t u x f x u f = scheme (9). If call such method of integration in time as Lax-type scheme, then one can use dierent numerical quadrature formulae for integration over x. For the midpoint or the trapezoidal rule for these integrals 8 dierent schemes are generated. Question: how close are properties of these 8 schemes? A partial answer can be obtained by using the dierential approximation technique.

33 Algorithmic Generation of Dierence Schemes In (G.,B.,Mozzhilkin'06) we suggested an algorithmic approach to generation of dierence schemes. By example consider equation (8). u t dt = u, u t = u n+1 j un j+2 + un j, 2 f x dx = f, = 2 h(f x ) n j+1 = fj+2 n fj n, (12) u x dx = u, 2 h(u x ) n j+1 = u n j+2 u n j, u xx dx = u x, 2 h(u xx ) n j+1 = (u x ) n j+2 (u x ) n j. Gr obner basis for u xx u t u x f x u f = scheme (9). If call such method of integration in time as Lax-type scheme, then one can use dierent numerical quadrature formulae for integration over x. For the midpoint or the trapezoidal rule for these integrals 8 dierent schemes are generated. Question: how close are properties of these 8 schemes? A partial answer can be obtained by using the dierential approximation technique.

34 Algorithm for Hyperbolic Form Since in advance the order of ratio h is not known, one cannot specify a linear oder for PDE in construction of the dierential approximation. Equation (10) multiplied by it can be partition in three groups: [ (ut + f x νu xx ) 1 2 u xxh 2, 1 2 u tt 2 + ( 1 6 f xxx 1 3 νu xxxx)h u xxxxh 4, 1 6 u ttt 3 + ( f xxxxx 2 45 νu xxxxxx)h u xxxxxxh 6] The rst group does not divisors. The second group has divisors in the rst group. The third group has divisors from the rst and the second groups. The truncation order for the Taylor expansion is specied in such a way in order to provide a correct partition into groups.

35 Algorithm for Hyperbolic Form Since in advance the order of ratio h is not known, one cannot specify a linear oder for PDE in construction of the dierential approximation. Equation (10) multiplied by it can be partition in three groups: [ (ut + f x νu xx ) 1 2 u xxh 2, 1 2 u tt 2 + ( 1 6 f xxx 1 3 νu xxxx)h u xxxxh 4, 1 6 u ttt 3 + ( f xxxxx 2 45 νu xxxxxx)h u xxxxxxh 6] The rst group does not divisors. The second group has divisors in the rst group. The third group has divisors from the rst and the second groups. The truncation order for the Taylor expansion is specied in such a way in order to provide a correct partition into groups.

36 Algorithm for Hyperbolic Form Since in advance the order of ratio h is not known, one cannot specify a linear oder for PDE in construction of the dierential approximation. Equation (10) multiplied by it can be partition in three groups: [ (ut + f x νu xx ) 1 2 u xxh 2, 1 2 u tt 2 + ( 1 6 f xxx 1 3 νu xxxx)h u xxxxh 4, 1 6 u ttt 3 + ( f xxxxx 2 45 νu xxxxxx)h u xxxxxxh 6] The rst group does not divisors. The second group has divisors in the rst group. The third group has divisors from the rst and the second groups. The truncation order for the Taylor expansion is specied in such a way in order to provide a correct partition into groups.

37 Algorithm for Hyperbolic Form To construct FDA for the parabolic form (it is sucient to compare scheme properties), the derivatives in t in the second group are replaced by their values from the rst group. It can be achieved by the sequential substitution derivatives u tt u tx u t in accordance with the lexicographic order: [ (u t + uu x νu xx ) u xx h 2 2, +( νuu xxx 2νu x u xx ν2 u xxxx u2 u xx + uu 2 x) 2 +( 1 6 u xxxxν 1 3 u xxxu 1 2 u xxu x )h u xxxxh 4]

38 Algorithm for Hyperbolic Form To construct FDA for the parabolic form (it is sucient to compare scheme properties), the derivatives in t in the second group are replaced by their values from the rst group. It can be achieved by the sequential substitution derivatives u tt u tx u t in accordance with the lexicographic order: [ (u t + uu x νu xx ) u xx h 2 2, +( νuu xxx 2νu x u xx ν2 u xxxx u2 u xx + uu 2 x) 2 +( 1 6 u xxxxν 1 3 u xxxu 1 2 u xxu x )h u xxxxh 4]

39 Implementation in Maple We implemented the above described method in Maple as a package FDA. >restart; >libname:=libname, /usr/local/lib/lfdm, /usr/local/lib/janet, /usr/local/lib/fda; /opt/maple10/lib, /usr/local/lib/lfdm, /usr/local/lib/janet, /usr/local/lib/fda >with(lfdm); [AEqn, AppShiftOp, AssertJanetBasis, CartanCharacter, CompCond, CompCondBasis, Di2Shift, FactorModuleBasis, HF, HP, HilbertFunction, HilbertPolynomial, HilbertSeries, IndexRegularity, InvReduce, JanetBasis, LFDMOptions, LeadingOp, Pol2Shift, Shift2Di, Shift2Op, Shift2Pol, ShiftGroebnerBasis, ShiftTabVar, WeightedHilbertSeries, ZeroSets] >with(fda); [DForm, PForm]

40 Implementation in Maple We implemented the above described method in Maple as a package FDA. >restart; >libname:=libname, /usr/local/lib/lfdm, /usr/local/lib/janet, /usr/local/lib/fda; /opt/maple10/lib, /usr/local/lib/lfdm, /usr/local/lib/janet, /usr/local/lib/fda >with(lfdm); [AEqn, AppShiftOp, AssertJanetBasis, CartanCharacter, CompCond, CompCondBasis, Di2Shift, FactorModuleBasis, HF, HP, HilbertFunction, HilbertPolynomial, HilbertSeries, IndexRegularity, InvReduce, JanetBasis, LFDMOptions, LeadingOp, Pol2Shift, Shift2Di, Shift2Op, Shift2Pol, ShiftGroebnerBasis, ShiftTabVar, WeightedHilbertSeries, ZeroSets] >with(fda); [DForm, PForm]

41 Implementation in Maple We implemented the above described method in Maple as a package FDA. >restart; >libname:=libname, /usr/local/lib/lfdm, /usr/local/lib/janet, /usr/local/lib/fda; /opt/maple10/lib, /usr/local/lib/lfdm, /usr/local/lib/janet, /usr/local/lib/fda >with(lfdm); [AEqn, AppShiftOp, AssertJanetBasis, CartanCharacter, CompCond, CompCondBasis, Di2Shift, FactorModuleBasis, HF, HP, HilbertFunction, HilbertPolynomial, HilbertSeries, IndexRegularity, InvReduce, JanetBasis, LFDMOptions, LeadingOp, Pol2Shift, Shift2Di, Shift2Op, Shift2Pol, ShiftGroebnerBasis, ShiftTabVar, WeightedHilbertSeries, ZeroSets] >with(fda); [DForm, PForm]

42 Implementation in Maple We implemented the above described method in Maple as a package FDA. >restart; >libname:=libname, /usr/local/lib/lfdm, /usr/local/lib/janet, /usr/local/lib/fda; /opt/maple10/lib, /usr/local/lib/lfdm, /usr/local/lib/janet, /usr/local/lib/fda >with(lfdm); [AEqn, AppShiftOp, AssertJanetBasis, CartanCharacter, CompCond, CompCondBasis, Di2Shift, FactorModuleBasis, HF, HP, HilbertFunction, HilbertPolynomial, HilbertSeries, IndexRegularity, InvReduce, JanetBasis, LFDMOptions, LeadingOp, Pol2Shift, Shift2Di, Shift2Op, Shift2Pol, ShiftGroebnerBasis, ShiftTabVar, WeightedHilbertSeries, ZeroSets] >with(fda); [DForm, PForm]

43 Implementation in Maple We implemented the above described method in Maple as a package FDA. >restart; >libname:=libname, /usr/local/lib/lfdm, /usr/local/lib/janet, /usr/local/lib/fda; /opt/maple10/lib, /usr/local/lib/lfdm, /usr/local/lib/janet, /usr/local/lib/fda >with(lfdm); [AEqn, AppShiftOp, AssertJanetBasis, CartanCharacter, CompCond, CompCondBasis, Di2Shift, FactorModuleBasis, HF, HP, HilbertFunction, HilbertPolynomial, HilbertSeries, IndexRegularity, InvReduce, JanetBasis, LFDMOptions, LeadingOp, Pol2Shift, Shift2Di, Shift2Op, Shift2Pol, ShiftGroebnerBasis, ShiftTabVar, WeightedHilbertSeries, ZeroSets] >with(fda); [DForm, PForm]

44 Implementation in Maple We implemented the above described method in Maple as a package FDA. >restart; >libname:=libname, /usr/local/lib/lfdm, /usr/local/lib/janet, /usr/local/lib/fda; /opt/maple10/lib, /usr/local/lib/lfdm, /usr/local/lib/janet, /usr/local/lib/fda >with(lfdm); [AEqn, AppShiftOp, AssertJanetBasis, CartanCharacter, CompCond, CompCondBasis, Di2Shift, FactorModuleBasis, HF, HP, HilbertFunction, HilbertPolynomial, HilbertSeries, IndexRegularity, InvReduce, JanetBasis, LFDMOptions, LeadingOp, Pol2Shift, Shift2Di, Shift2Op, Shift2Pol, ShiftGroebnerBasis, ShiftTabVar, WeightedHilbertSeries, ZeroSets] >with(fda); [DForm, PForm]

45 Example: Burgers' Equation >L:=[ut(n,j)+Fx(n,j)-nu*uxx(n,j), > ut(n,j+1)*tau-(u(n+1,j+1)-(u(n,j+2)+u(n,j))/2), > 2*Fx(n,j+1)*h-(F(n,j+2)-F(n,j)), > 2*ux(n,j+1)*h-(u(n,j+2)-u(n,j)), > 2*uxx(n,j+1)*h-(ux(n,j+2)-ux(n,j))]; [ut (n, j) + Fx (n, j) ν uxx (n, j), ut (n, j + 1) u (n + 1, j + 1) + 1 u (n, j + 2) + 1 u (n, j), Fx (n, j + 1) h F (n, j + 2) + F (n, j), 2 ux (n, j + 1) h u (n, j + 2) + u (n, j), 2 uxx (n, j + 1) h ux (n, j + 2) + ux (n, j)] >JanetBasis(L, [n,j], [uxx,ux,ut,fx,u,f],2); [[ 2 ν u (n, j + 2) + ν u (n, j) + ν u (n, j + 4) 4 h 2 u (n + 1, j + 2) + 2 h 2 u (n, j + 3) + 2 h 2 u (n, j + 1) 2 hf (n, j + 3) + 2 hf (n, j + 1), 2 Fx (n, j + 1) h F (n, j + 2) + F (n, j), 2 ut (n, j + 1) 2 u (n + 1, j + 1) + u (n, j + 2) + u (n, j), 2 hν ux (n, j) ν u (n, j + 3) + ν u (n, j + 1) + 4 h 2 u (n + 1, j + 1) 2 h 2 u (n, j + 2) 2 h 2 u (n, j) + 2 hf (n, j + 2) 2 hf (n, j), ut (n, j) + Fx (n, j) ν uxx (n, j)], [n, j], [uxx, ux, ut, Fx, u, F]]

46 Example: Burgers' Equation >L:=[ut(n,j)+Fx(n,j)-nu*uxx(n,j), > ut(n,j+1)*tau-(u(n+1,j+1)-(u(n,j+2)+u(n,j))/2), > 2*Fx(n,j+1)*h-(F(n,j+2)-F(n,j)), > 2*ux(n,j+1)*h-(u(n,j+2)-u(n,j)), > 2*uxx(n,j+1)*h-(ux(n,j+2)-ux(n,j))]; [ut (n, j) + Fx (n, j) ν uxx (n, j), ut (n, j + 1) u (n + 1, j + 1) + 1 u (n, j + 2) + 1 u (n, j), Fx (n, j + 1) h F (n, j + 2) + F (n, j), 2 ux (n, j + 1) h u (n, j + 2) + u (n, j), 2 uxx (n, j + 1) h ux (n, j + 2) + ux (n, j)] >JanetBasis(L, [n,j], [uxx,ux,ut,fx,u,f],2); [[ 2 ν u (n, j + 2) + ν u (n, j) + ν u (n, j + 4) 4 h 2 u (n + 1, j + 2) + 2 h 2 u (n, j + 3) + 2 h 2 u (n, j + 1) 2 hf (n, j + 3) + 2 hf (n, j + 1), 2 Fx (n, j + 1) h F (n, j + 2) + F (n, j), 2 ut (n, j + 1) 2 u (n + 1, j + 1) + u (n, j + 2) + u (n, j), 2 hν ux (n, j) ν u (n, j + 3) + ν u (n, j + 1) + 4 h 2 u (n + 1, j + 1) 2 h 2 u (n, j + 2) 2 h 2 u (n, j) + 2 hf (n, j + 2) 2 hf (n, j), ut (n, j) + Fx (n, j) ν uxx (n, j)], [n, j], [uxx, ux, ut, Fx, u, F]]

47 Example: Burgers' Equation >L:=[ut(n,j)+Fx(n,j)-nu*uxx(n,j), > ut(n,j+1)*tau-(u(n+1,j+1)-(u(n,j+2)+u(n,j))/2), > 2*Fx(n,j+1)*h-(F(n,j+2)-F(n,j)), > 2*ux(n,j+1)*h-(u(n,j+2)-u(n,j)), > 2*uxx(n,j+1)*h-(ux(n,j+2)-ux(n,j))]; [ut (n, j) + Fx (n, j) ν uxx (n, j), ut (n, j + 1) u (n + 1, j + 1) + 1 u (n, j + 2) + 1 u (n, j), Fx (n, j + 1) h F (n, j + 2) + F (n, j), 2 ux (n, j + 1) h u (n, j + 2) + u (n, j), 2 uxx (n, j + 1) h ux (n, j + 2) + ux (n, j)] >JanetBasis(L, [n,j], [uxx,ux,ut,fx,u,f],2); [[ 2 ν u (n, j + 2) + ν u (n, j) + ν u (n, j + 4) 4 h 2 u (n + 1, j + 2) + 2 h 2 u (n, j + 3) + 2 h 2 u (n, j + 1) 2 hf (n, j + 3) + 2 hf (n, j + 1), 2 Fx (n, j + 1) h F (n, j + 2) + F (n, j), 2 ut (n, j + 1) 2 u (n + 1, j + 1) + u (n, j + 2) + u (n, j), 2 hν ux (n, j) ν u (n, j + 3) + ν u (n, j + 1) + 4 h 2 u (n + 1, j + 1) 2 h 2 u (n, j + 2) 2 h 2 u (n, j) + 2 hf (n, j + 2) 2 hf (n, j), ut (n, j) + Fx (n, j) ν uxx (n, j)], [n, j], [uxx, ux, ut, Fx, u, F]]

48 Example: Burgers' Equation >L:=[ut(n,j)+Fx(n,j)-nu*uxx(n,j), > ut(n,j+1)*tau-(u(n+1,j+1)-(u(n,j+2)+u(n,j))/2), > 2*Fx(n,j+1)*h-(F(n,j+2)-F(n,j)), > 2*ux(n,j+1)*h-(u(n,j+2)-u(n,j)), > 2*uxx(n,j+1)*h-(ux(n,j+2)-ux(n,j))]; [ut (n, j) + Fx (n, j) ν uxx (n, j), ut (n, j + 1) u (n + 1, j + 1) + 1 u (n, j + 2) + 1 u (n, j), Fx (n, j + 1) h F (n, j + 2) + F (n, j), 2 ux (n, j + 1) h u (n, j + 2) + u (n, j), 2 uxx (n, j + 1) h ux (n, j + 2) + ux (n, j)] >JanetBasis(L, [n,j], [uxx,ux,ut,fx,u,f],2); [[ 2 ν u (n, j + 2) + ν u (n, j) + ν u (n, j + 4) 4 h 2 u (n + 1, j + 2) + 2 h 2 u (n, j + 3) + 2 h 2 u (n, j + 1) 2 hf (n, j + 3) + 2 hf (n, j + 1), 2 Fx (n, j + 1) h F (n, j + 2) + F (n, j), 2 ut (n, j + 1) 2 u (n + 1, j + 1) + u (n, j + 2) + u (n, j), 2 hν ux (n, j) ν u (n, j + 3) + ν u (n, j + 1) + 4 h 2 u (n + 1, j + 1) 2 h 2 u (n, j + 2) 2 h 2 u (n, j) + 2 hf (n, j + 2) 2 hf (n, j), ut (n, j) + Fx (n, j) ν uxx (n, j)], [n, j], [uxx, ux, ut, Fx, u, F]]

49 >collect(%[1,1]/(4*tau*h^2),[tau,h,nu]); 1/2 F (n, j + 3) + 1/2 F (n, j + 1) (1/4 u (n, j + 4) 1/2 u (n, j + 2) + 1/4 u (n, j)) ν + h h 2 1/2 u (n, j + 3) + 1/2 u (n, j + 1) u (n + 1, j + 2) + >a:=-dform(%,[u,f],[[n,tau,t],[j,h,x]],[0,2],2); [D 2 (F) (t, x) D 2,2 (u) (t, x) ν + D 1 (u) (t, x) 1/2 D 2,2 (u) (t, x) h 2, 1/2 D 1,1 (u) (t, x) ( 1/6 D 2,2,2 (F) (t, x) + 1/3 D 2,2,2,2 (u) (t, x)) νh 2 1/24 D 2,2,2,2 (u) (t, x) h 4, 1/6 D 1,1,1 (u) (t, x) D 2,2,2,2,2 (F) (t, x) D 2,2,2,2,2,2 (u) (t, x) νh 4 1 D 2,2,2,2,2,2 (u) (t, x) h 6 ] 720

50 >collect(%[1,1]/(4*tau*h^2),[tau,h,nu]); 1/2 F (n, j + 3) + 1/2 F (n, j + 1) (1/4 u (n, j + 4) 1/2 u (n, j + 2) + 1/4 u (n, j)) ν + h h 2 1/2 u (n, j + 3) + 1/2 u (n, j + 1) u (n + 1, j + 2) + >a:=-dform(%,[u,f],[[n,tau,t],[j,h,x]],[0,2],2); [D 2 (F) (t, x) D 2,2 (u) (t, x) ν + D 1 (u) (t, x) 1/2 D 2,2 (u) (t, x) h 2, 1/2 D 1,1 (u) (t, x) ( 1/6 D 2,2,2 (F) (t, x) + 1/3 D 2,2,2,2 (u) (t, x)) νh 2 1/24 D 2,2,2,2 (u) (t, x) h 4, 1/6 D 1,1,1 (u) (t, x) D 2,2,2,2,2 (F) (t, x) D 2,2,2,2,2,2 (u) (t, x) νh 4 1 D 2,2,2,2,2,2 (u) (t, x) h 6 ] 720

51 >collect(%[1,1]/(4*tau*h^2),[tau,h,nu]); 1/2 F (n, j + 3) + 1/2 F (n, j + 1) (1/4 u (n, j + 4) 1/2 u (n, j + 2) + 1/4 u (n, j)) ν + h h 2 1/2 u (n, j + 3) + 1/2 u (n, j + 1) u (n + 1, j + 2) + >a:=-dform(%,[u,f],[[n,tau,t],[j,h,x]],[0,2],2); [D 2 (F) (t, x) D 2,2 (u) (t, x) ν + D 1 (u) (t, x) 1/2 D 2,2 (u) (t, x) h 2, 1/2 D 1,1 (u) (t, x) ( 1/6 D 2,2,2 (F) (t, x) + 1/3 D 2,2,2,2 (u) (t, x)) νh 2 1/24 D 2,2,2,2 (u) (t, x) h 4, 1/6 D 1,1,1 (u) (t, x) D 2,2,2,2,2 (F) (t, x) D 2,2,2,2,2,2 (u) (t, x) νh 4 1 D 2,2,2,2,2,2 (u) (t, x) h 6 ] 720

52 >F:=u^2/2: >PForm(a); [D 2 (u) (t, x) u (t, x) D 2,2 (u) (t, x) ν + D 1 (u) (t, x) 1/2 D 2,2 (u) (t, x) h 2, ( ν D 2,2,2 (u) (t, x) u (t, x) 2 ν D 2,2 (u) (t, x) D 2 (u) (t, x) +1/2 ν 2 D 2,2,2,2 (u) (t, x) + 1/2 (u (t, x)) 2 D 2,2 (u) (t, x) + u (t, x) (D 2 (u) (t, x)) 2 + ( 1/3 D 2,2,2 (u) (t, x) u (t, x) 1/2 D 2,2 (u) (t, x) D 2 (u) (t, x) + 1/6 D 2,2,2,2 (u) (t, x) ν) h 2 + 1/12 D 2,2,2,2 (u) (t, x) h 4 ]

53 >F:=u^2/2: >PForm(a); [D 2 (u) (t, x) u (t, x) D 2,2 (u) (t, x) ν + D 1 (u) (t, x) 1/2 D 2,2 (u) (t, x) h 2, ( ν D 2,2,2 (u) (t, x) u (t, x) 2 ν D 2,2 (u) (t, x) D 2 (u) (t, x) +1/2 ν 2 D 2,2,2,2 (u) (t, x) + 1/2 (u (t, x)) 2 D 2,2 (u) (t, x) + u (t, x) (D 2 (u) (t, x)) 2 + ( 1/3 D 2,2,2 (u) (t, x) u (t, x) 1/2 D 2,2 (u) (t, x) D 2 (u) (t, x) + 1/6 D 2,2,2,2 (u) (t, x) ν) h 2 + 1/12 D 2,2,2,2 (u) (t, x) h 4 ]

54 One can also use the trapezoidal rule for spatial integration. This derives other schemes. By selecting either the midpoint or the trapezoidal rule for the spatial integrals, we obtain 8 possible schemes. Among them there are 7 dierent schemes: 2(u n+1 j+2 + un+1 j+1 ) (un j+3 + un j+2 + un j+1 + un j ) 4 2u n+1 j+1 (un j+2 + un j ) 2 (f n j+3 + fn j+2 ) (fn j+1 + fn j ) + = ν 4h + f n j+2 fn j 2h = ν u n j+2 2un j+1 + un j h 2, 2(u n+1 j+3 + 2un+1 j+2 + un+1 j+1 ) (un j+4 + 2un j+3 + 2un j+2 + 2un j+1 + un j ) 8 (f n j+4 + 2fn j+1 ) (2fn j+1 + fn j ) u n j+3 2un j+2 + un j+1 + = ν, 8h h 2 2(u n+1 j+3 + un+1 j+2 ) (un j+4 + un j+3 + un j+2 + un j+1 ) 4 2(u n+1 j+2 + un+1 j+1 ) (un j+3 + un j+2 + un j+1 + un j ) 4 + f n j+3 fn j+2 ((u n j+5 + un j+4 ) 2un j+3 ) (2un j+2 (un j+1 + un j )) = ν, 8h 2 + f n j+2 fn j+1 h = ν 2(u n+1 j+3 + 2un+1 j+2 + un+1 j+1 ) (un j+4 + 2un j+3 + 2un j+2 + 2un j+1 + un j ) 8 = ν u n j+3 2un j+2 + un j+1 h 2. h (u n j+3 un j+2 ) (un j+1 un j ) 2h 2, (u n j+3 un j+2 ) (un j+1 un j ) 2h 2, + f n j+3 fn j+1 2h

55 Computation of dierential approximations for them gives: (9) [..., (...) +( 1 uxxxxν 1 uxxxu uxxux)h uxxxx h 4 (??) [..., (...) +( 1 uxxxxν 1 uxxxu 2 2 uxxux)h uxxxx h 4 (??) [..., (...) +( 1 uxxxxν 1 uxxxu 2 2 uxxux)h uxxxx h 4 (??) [..., (...) +( 2 uxxxxν 1 uxxxu uxxux)h uxxxx h 4 (??) [..., (...) +( 1 uxxxxν 1 uxxxu uxxux)h uxxxx h 4 (??) [..., (...) +( 1 uxxxxν 1 uxxxu 2 2 uxxux)h uxxxx h 4 (??) [..., (...) +( 2 uxxxxν 7 uxxxu uxxux)h uxxxx h 4 (13) These schemes have similar properties, and three of them have identical dierential approximations. By inspection of the schemes we see that these schemes have the same order of approximations; identical dissipative properties; very close dispersion properties with some small distinctions in the rational coecients at derivatives in the factors at h 2 and h 4 /.

56 Computation of dierential approximations for them gives: (9) [..., (...) +( 1 uxxxxν 1 uxxxu uxxux)h uxxxx h 4 (??) [..., (...) +( 1 uxxxxν 1 uxxxu 2 2 uxxux)h uxxxx h 4 (??) [..., (...) +( 1 uxxxxν 1 uxxxu 2 2 uxxux)h uxxxx h 4 (??) [..., (...) +( 2 uxxxxν 1 uxxxu uxxux)h uxxxx h 4 (??) [..., (...) +( 1 uxxxxν 1 uxxxu uxxux)h uxxxx h 4 (??) [..., (...) +( 1 uxxxxν 1 uxxxu 2 2 uxxux)h uxxxx h 4 (??) [..., (...) +( 2 uxxxxν 7 uxxxu uxxux)h uxxxx h 4 (13) These schemes have similar properties, and three of them have identical dierential approximations. By inspection of the schemes we see that these schemes have the same order of approximations; identical dissipative properties; very close dispersion properties with some small distinctions in the rational coecients at derivatives in the factors at h 2 and h 4 /.

57 Computation of dierential approximations for them gives: (9) [..., (...) +( 1 uxxxxν 1 uxxxu uxxux)h uxxxx h 4 (??) [..., (...) +( 1 uxxxxν 1 uxxxu 2 2 uxxux)h uxxxx h 4 (??) [..., (...) +( 1 uxxxxν 1 uxxxu 2 2 uxxux)h uxxxx h 4 (??) [..., (...) +( 2 uxxxxν 1 uxxxu uxxux)h uxxxx h 4 (??) [..., (...) +( 1 uxxxxν 1 uxxxu uxxux)h uxxxx h 4 (??) [..., (...) +( 1 uxxxxν 1 uxxxu 2 2 uxxux)h uxxxx h 4 (??) [..., (...) +( 2 uxxxxν 7 uxxxu uxxux)h uxxxx h 4 (13) These schemes have similar properties, and three of them have identical dierential approximations. By inspection of the schemes we see that these schemes have the same order of approximations; identical dissipative properties; very close dispersion properties with some small distinctions in the rational coecients at derivatives in the factors at h 2 and h 4 /.

58 Computation of dierential approximations for them gives: (9) [..., (...) +( 1 uxxxxν 1 uxxxu uxxux)h uxxxx h 4 (??) [..., (...) +( 1 uxxxxν 1 uxxxu 2 2 uxxux)h uxxxx h 4 (??) [..., (...) +( 1 uxxxxν 1 uxxxu 2 2 uxxux)h uxxxx h 4 (??) [..., (...) +( 2 uxxxxν 1 uxxxu uxxux)h uxxxx h 4 (??) [..., (...) +( 1 uxxxxν 1 uxxxu uxxux)h uxxxx h 4 (??) [..., (...) +( 1 uxxxxν 1 uxxxu 2 2 uxxux)h uxxxx h 4 (??) [..., (...) +( 2 uxxxxν 7 uxxxu uxxux)h uxxxx h 4 (13) These schemes have similar properties, and three of them have identical dierential approximations. By inspection of the schemes we see that these schemes have the same order of approximations; identical dissipative properties; very close dispersion properties with some small distinctions in the rational coecients at derivatives in the factors at h 2 and h 4 /.

59 Computation of dierential approximations for them gives: (9) [..., (...) +( 1 uxxxxν 1 uxxxu uxxux)h uxxxx h 4 (??) [..., (...) +( 1 uxxxxν 1 uxxxu 2 2 uxxux)h uxxxx h 4 (??) [..., (...) +( 1 uxxxxν 1 uxxxu 2 2 uxxux)h uxxxx h 4 (??) [..., (...) +( 2 uxxxxν 1 uxxxu uxxux)h uxxxx h 4 (??) [..., (...) +( 1 uxxxxν 1 uxxxu uxxux)h uxxxx h 4 (??) [..., (...) +( 1 uxxxxν 1 uxxxu 2 2 uxxux)h uxxxx h 4 (??) [..., (...) +( 2 uxxxxν 7 uxxxu uxxux)h uxxxx h 4 (13) These schemes have similar properties, and three of them have identical dierential approximations. By inspection of the schemes we see that these schemes have the same order of approximations; identical dissipative properties; very close dispersion properties with some small distinctions in the rational coecients at derivatives in the factors at h 2 and h 4 /.

60 Denoting the values of functions on the intermediate time level by u, F we obtain the following dierence system: n n n u t j + F x j = ν u xx j u t n j = u n+1 j un j+2 +un j 2 2F x n j+1 h = F n j+2 Fn j 2u x n j+1 h = u n j+2 un j 2u xx n j+1 h = u x n j+2 u x n j u t n j + F x n j = ν u xx n j n u t j = u n+1 j u n j n 2F x j+1 h = Fn j+2 F n j n 2u x j+1 h = u n j+2 u n j 2u x x n j+1 h = u x n n j+2 u x j. The trapezoidal or midpoint rule for the integral relation between u x and u yields 36 dierent schemes whose are similar the Lax-Wendro scheme. (14)

61 Conclusions Gr obner bases provide algorithmic construction of nite dierence schemes for linear PDEs in two independent variables. Having a dierence scheme constructed the method of dierential approximation (modied equation) allows to study stability of schemes for a wide class of PDEs. In particular, the rst dierential approximation (FDA) plays an important role in the stability analysis. For linear and some quasilinear PDEs dierential approximations can be constructed algorithmically, and the underlying algorithms have been implemented in Maple. Algorithms for computing parabolic and hyperbolic forms of FDA are available together with their implementation in Maple. The methods and software designed were applied to many dierent PDEs, for example, to Burgers' equation. A number of dierence schemes for them was generated and their stability properties were studied by the method of dierential approximation.

62 Conclusions Gr obner bases provide algorithmic construction of nite dierence schemes for linear PDEs in two independent variables. Having a dierence scheme constructed the method of dierential approximation (modied equation) allows to study stability of schemes for a wide class of PDEs. In particular, the rst dierential approximation (FDA) plays an important role in the stability analysis. For linear and some quasilinear PDEs dierential approximations can be constructed algorithmically, and the underlying algorithms have been implemented in Maple. Algorithms for computing parabolic and hyperbolic forms of FDA are available together with their implementation in Maple. The methods and software designed were applied to many dierent PDEs, for example, to Burgers' equation. A number of dierence schemes for them was generated and their stability properties were studied by the method of dierential approximation.

63 Conclusions Gr obner bases provide algorithmic construction of nite dierence schemes for linear PDEs in two independent variables. Having a dierence scheme constructed the method of dierential approximation (modied equation) allows to study stability of schemes for a wide class of PDEs. In particular, the rst dierential approximation (FDA) plays an important role in the stability analysis. For linear and some quasilinear PDEs dierential approximations can be constructed algorithmically, and the underlying algorithms have been implemented in Maple. Algorithms for computing parabolic and hyperbolic forms of FDA are available together with their implementation in Maple. The methods and software designed were applied to many dierent PDEs, for example, to Burgers' equation. A number of dierence schemes for them was generated and their stability properties were studied by the method of dierential approximation.

64 Conclusions Gr obner bases provide algorithmic construction of nite dierence schemes for linear PDEs in two independent variables. Having a dierence scheme constructed the method of dierential approximation (modied equation) allows to study stability of schemes for a wide class of PDEs. In particular, the rst dierential approximation (FDA) plays an important role in the stability analysis. For linear and some quasilinear PDEs dierential approximations can be constructed algorithmically, and the underlying algorithms have been implemented in Maple. Algorithms for computing parabolic and hyperbolic forms of FDA are available together with their implementation in Maple. The methods and software designed were applied to many dierent PDEs, for example, to Burgers' equation. A number of dierence schemes for them was generated and their stability properties were studied by the method of dierential approximation.

65 Conclusions Gr obner bases provide algorithmic construction of nite dierence schemes for linear PDEs in two independent variables. Having a dierence scheme constructed the method of dierential approximation (modied equation) allows to study stability of schemes for a wide class of PDEs. In particular, the rst dierential approximation (FDA) plays an important role in the stability analysis. For linear and some quasilinear PDEs dierential approximations can be constructed algorithmically, and the underlying algorithms have been implemented in Maple. Algorithms for computing parabolic and hyperbolic forms of FDA are available together with their implementation in Maple. The methods and software designed were applied to many dierent PDEs, for example, to Burgers' equation. A number of dierence schemes for them was generated and their stability properties were studied by the method of dierential approximation.

66 References Gerdt V.P. and Blinkov Yu. A. and Mozzhilkin V. V. Gr obner bases and generation of dierence schemes for partial dierential equations. Symmetry, Integrability and Geometry: Methods and Applications, 2:26, Ganzha V.G. and Vorozhtsov E.V. Computer-aided analysis of dierence schemes for partial dierential equations. New York, Wiley-Interscience, Strikwerda J.C. Finite dierence schemes and partial dierential equations. Philadelphia, SIAM, Shokin Yu.I. and Yanenko N.N. Method of dierential approximation. Application to gas dynamics. Nauka, Siberian Division, 1985 (in Russian). Gerdt V.P. and Robertz D. A Maple package for computing Gr obner bases for linear recurrence relations. Nuclear Instruments and Methods in Physics Research, A559:215219, 2006.

67 References Gerdt V.P. and Blinkov Yu. A. and Mozzhilkin V. V. Gr obner bases and generation of dierence schemes for partial dierential equations. Symmetry, Integrability and Geometry: Methods and Applications, 2:26, Ganzha V.G. and Vorozhtsov E.V. Computer-aided analysis of dierence schemes for partial dierential equations. New York, Wiley-Interscience, Strikwerda J.C. Finite dierence schemes and partial dierential equations. Philadelphia, SIAM, Shokin Yu.I. and Yanenko N.N. Method of dierential approximation. Application to gas dynamics. Nauka, Siberian Division, 1985 (in Russian). Gerdt V.P. and Robertz D. A Maple package for computing Gr obner bases for linear recurrence relations. Nuclear Instruments and Methods in Physics Research, A559:215219, 2006.

68 References Gerdt V.P. and Blinkov Yu. A. and Mozzhilkin V. V. Gr obner bases and generation of dierence schemes for partial dierential equations. Symmetry, Integrability and Geometry: Methods and Applications, 2:26, Ganzha V.G. and Vorozhtsov E.V. Computer-aided analysis of dierence schemes for partial dierential equations. New York, Wiley-Interscience, Strikwerda J.C. Finite dierence schemes and partial dierential equations. Philadelphia, SIAM, Shokin Yu.I. and Yanenko N.N. Method of dierential approximation. Application to gas dynamics. Nauka, Siberian Division, 1985 (in Russian). Gerdt V.P. and Robertz D. A Maple package for computing Gr obner bases for linear recurrence relations. Nuclear Instruments and Methods in Physics Research, A559:215219, 2006.

69 References Gerdt V.P. and Blinkov Yu. A. and Mozzhilkin V. V. Gr obner bases and generation of dierence schemes for partial dierential equations. Symmetry, Integrability and Geometry: Methods and Applications, 2:26, Ganzha V.G. and Vorozhtsov E.V. Computer-aided analysis of dierence schemes for partial dierential equations. New York, Wiley-Interscience, Strikwerda J.C. Finite dierence schemes and partial dierential equations. Philadelphia, SIAM, Shokin Yu.I. and Yanenko N.N. Method of dierential approximation. Application to gas dynamics. Nauka, Siberian Division, 1985 (in Russian). Gerdt V.P. and Robertz D. A Maple package for computing Gr obner bases for linear recurrence relations. Nuclear Instruments and Methods in Physics Research, A559:215219, 2006.

70 References Gerdt V.P. and Blinkov Yu. A. and Mozzhilkin V. V. Gr obner bases and generation of dierence schemes for partial dierential equations. Symmetry, Integrability and Geometry: Methods and Applications, 2:26, Ganzha V.G. and Vorozhtsov E.V. Computer-aided analysis of dierence schemes for partial dierential equations. New York, Wiley-Interscience, Strikwerda J.C. Finite dierence schemes and partial dierential equations. Philadelphia, SIAM, Shokin Yu.I. and Yanenko N.N. Method of dierential approximation. Application to gas dynamics. Nauka, Siberian Division, 1985 (in Russian). Gerdt V.P. and Robertz D. A Maple package for computing Gr obner bases for linear recurrence relations. Nuclear Instruments and Methods in Physics Research, A559:215219, 2006.