1 Simple Euler method and its modifications

Transcription

1 MATH 337, by T. Lakoba, University of Vermont 6 1 Simple Euler metod and its modifications 1.1 Simple Euler metod for te 1st-order IVP Consider te IVP Let: x i = x 0 + i, y i = yx i ) Y i y x) = fx, y), yx 0 ) = y ) i = 0, 1,... n true solution evaluated at points x i te solution to be calculated numerically. Replace Ten Eq. 1.1) gets replaced wit y x) Y i+1 Y i. Y i+1 = Y i + fx i, Y i ) Y 0 = y 0. 1.) 1. Local error of te simple Euler metod Te calculated solution satisfies Eq. 1.). Next, assuming tat te true solution of IVP 1.1) as at least) a second derivative y x), one can use te Taylor expansion to write: y i+1 = yx i + ) = y i + y i + y i x i ) = y i + fx i, y i ) + O ). 1.3) Here x i is some point between x i and x i+1 = x i +, and we ave used Eq. 0.5). Notation O k ) for any k means te following: For example, q = O k q ) wenever lim = const <, const 0. 0 k = O ); or 1 + cos3 + ) = O). We now introduce a new notation. Te local truncation error sows ow well te solution Y i+1 of te finite-difference sceme approximates te exact solution y i+1 of te ODE at point x i+1, assuming tat at x i te two solutions were te same, i.e. Y i = y i. Comparing te last line of Eq. 1.3) wit Eq. 1.), we see tat te local truncation error of te simple Euler metod is O ). It tends to zero wen 0. Anoter useful notation is tat of discretization error. It sows ow well te finite-difference sceme approximates te ODE. Let us now estimate tis error. First, we note from 1.) and 1.3) tat te computed and exact solutions satisfy: Y i+1 Y i = fx i, Y i ) and y i+1 y i = fx i, y i ) + O), wence te discretization error of te simple Euler metod is seen to be O).

2 MATH 337, by T. Lakoba, University of Vermont Global error of te Euler metod; Propagation of errors As we ave said above, te local truncation error sows ow well te computed solution approximates te exact solution at one given point, assuming tat tese two solutions ave been te same up to tat point. However, as we compute te solution of te finite-difference sceme, te local truncation errors at eac step accumulate. Tis results in tat te difference between te computed solution Y i and exact solution y i at some point x i down te line becomes muc greater tan te local truncation error. Let ɛ i = y i Y i denote te error te difference between te true and computed solutions) at x = x i. Tis error or, sometimes, its absolute value) is called te global error of te finite-difference metod. Our goal in tis subsection will be to find an upper bound for tis error. Let us empasize tat finding an upper bound for te error rater tan te error itself is te best one can do. Indeed, if one could ave found te actual error ɛ i, one would ave ten simply added it to te numerical solution Y i and obtained te exact solution y i.) Te main purpose of finding te upper bound for te error is to determine ow it depends on te step size. We will do tis now for te simple Euler metod 1.). To tis end, we begin by considering Eq. 1.) and te 1st line of Eq. 1.3): Y i+1 = Y i + fx i, Y i ) y i+1 = y i + fx i, y i ) + y x i ) Subtract te 1st equation above from te nd to obtain te error at x i+1 : ɛ i+1 = ɛ i + fx i, y i ) fx i, Y i )) + y x i ). 1.4) Now apply te triangle inequality, valid for any tree numbers a, b, c: to Eq. 1.4) and obtain: a = b + c a b + c, 1.5) ɛ i+1 ɛ i + L ɛ i + y x i ) = 1 + L) ɛ i + y x i ). 1.6) In writing te second term in te above formula, we used te fact tat fx, y) satisfies te Lipscitz condition wit respect to y see Lecture 0). To complete finding te upper bound for te error ɛ i+1, we need to estimate y x i ). We use te Cain rule for a function of two variables recall Calculus III) to obtain: y x) = d yx) dx use te ODE = dfx, y) dx Considering te first term on te r..s. of 1.7), let us assume tat dx = f x dx + f dy y dx = f x + f y f. 1.7) f x M 1 for some M 1 <. 1.8) In cases wen tis asumption does not old as, for example, for fx, y) = x 1/3 sin 1 x ), te estimate obtained below see 1.16)) is not valid, but a modified estimate can usually be found on a case-by-case basis. So ere we proceed wit assumption 1.8).

3 MATH 337, by T. Lakoba, University of Vermont 8 Considering te second term on te r..s. of 1.7), we first recall tat f satisfies te Lipscitz condition wit respect to y, wic means tat f y M for some M <, 1.9) except possibly at a finite number of y-values were f y does not exist like at y = 0 for fy) = y ). Finally, te oter factor of te second term on te r..s. of 1.7) is also bounded, because f is assumed to be continuous and on te closed region R see te Existence and Uniqueness Teorem in Lecture 0). Tus, Combining Eqs ), we see tat Now combining Eqs. 1.6) and 1.11), we obtain: f M 3 for some M 3 <. 1.10) y x i ) M 1 + M M 3 M <. 1.11) ɛ i L) ɛ i + M. 1.1) Tis last equation implies tat ɛ i+1 z i+1, were z i+1 satisfies te following recurrence equation: z i+1 = 1 + L)z i + M, z 0 = ) Condition z 0 = 0 follows from te fact tat ɛ 0 = 0; see te initial conditions in Eqs. 1.1) and 1.).) Tus, te error ɛ i is bounded by z i, and we need to solve Eq. 1.13) to find tat bound. Te way to do so is analogous to solving a linear inomogeneous equation see Section 0.4). However, before we obtain te solution, let us develop an intuitive understanding of wat kind of answer we sould expect. To tis end, let us assume for te moment tat L = 0 in Eq. 1.13). Ten we ave: z i+1 = z i + M = z i 1 + M) + M =... = z 0 + M i = 0 + M xi x 0 = Mx i x 0 ) Tat is, te global error ɛ i sould ave te size O). In oter words, = O). Global error = Number of steps Local error or O) = O 1 ) O ) Now let us sow tat a similar estimate also olds for L 0. First, solve te omogeneous version of 1.13): z i+1 = 1 + L) z i z i,om = 1 + L) i. 1.14) Note tat tis is an analogue of e ax i x 0 ) in Section 0.4, because 1 + L) i = 1 + L) x i x 0 )/ 0 e Lx i x 0 ),

4 MATH 337, by T. Lakoba, University of Vermont 9 were we ave used te definition of x i, found after 1.1), and also te results of Section 0.5. In close analogy to te metod used in Section 0.4, we seek te solution of 1.13) in te form z i = c i z i,om wit c 0 = 0). Substituting tis form into 1.13) and using Eq. 1.14), we obtain: c i L) i+1 = 1 + L) c i 1 + L) i + M c i+1 = c i + =... = c 0 + M 1 + L) = c M i+1 i L) + M i L) i+1 i+1 k=1 = geometric series M 1 + L) M L) k L) i L) = M L Combining 1.14) and 1.15), and using 0.16), we finally obtain: z i+1 = M L 1 + L) i+1 1 ) = M L 1 + L) x i+1 x 0 )/ 1 ) M L ) ) 1 + L) i+1 e Lx i+1 x 0 ) 1 ) = O), ɛ i+1 M e Lx x 0 ) 1 ) = O). 1.16) L Tis is te upper bound for te global error of te simple Euler metod 1.). Tus, in te last two subsections, we ave sown tat for te simple Euler metod: Local truncation error = O ); Discretization error = O); Global error = O). Te exponent of in te global error is often referred to as te order of te finite-difference metod. Tus, te simpler Euler metod is te 1st-order metod. Question: How does te above bound for te error cange wen we include te macine round-off error wic occurs because numbers are computed wit finite accuracy, usually )? Answer: In te above formulae, replace M/ by M/ + r, were r is te maximum value of te round-off error. Ten Eq. 1.16) gets replaced wit M ɛ i+1 ) 1 + r e Lx x 0 ) 1 ) = L M L + r ) e Lx x 0 ) 1 ) 1.17) L Te r..s. of te above bound is scematically plotted in te figure below. We see tat for very small, te term r/ can be dominant.

5 MATH 337, by T. Lakoba, University of Vermont 10 Moral: total error Decreasing te step size of te difference equation does not always result in te increased accuracy of te obtained solution. step size discretization error round off error 1.4 Modifications of te Euler metod In tis subsection, our goal is to find finite-difference scemes wic are more accurate tan te simple Euler metod i.e., te global error of te sougt metods sould be O ) or better). Again, we first want to develop an intuitive understanding of ow tis can be done, and ten actually do it. So, to begin, we notice an obvious fact tat te ODE y = fx, y) is just a more general case of y = fx). Te solution of te latter equation is y = fx)dx. Wenever we cannot evaluate te integral analytically in closed form, we resort to approximating te integral by te Riemann sums. A very crude approximation to b fx)dx a is provided by te left Riemann sums: left Riemann sums Y i+1 = Y i + fx i ). Tis is te analogue of te simple Euler metod 1.): Y i+1 = Y i + fx i, Y i ). x 0 x 1 x x 3 Approximations of te integral b a fx)dx tat are known to be more accurate tan te left Riemann sums are te Trapezoidal Rule and te Midpoint Rule:

6 MATH 337, by T. Lakoba, University of Vermont 11 Trapezoidal Rule: Y i+1 = Y i + fx i) + fx i+1 ). Its analogue for te ODE is to look like tis: Trapezoidal Rule Y i+1 = Y i + fx i, Y i ) + fx i+1, Y i + A)), 1.18) were te coefficient A is to be determined. Metod 1.18) is called te Modified Euler metod. Midpoint Rule: Y i+1 = Y i + f x i + ). x 0 x 1 x x 3 Midpoint Rule Its analogue for te ODE is to look like tis: Y i+1 = Y i + f x i + ), Y i + B, 1.19) were te coefficient B is to be determined. We will refer to metod 1.19) as te Midpoint metod. x 0 x 1 x x 3 Te coefficients A in 1.18) and B in 1.19) are determined from te requirement tat te corresponding finite-difference sceme ave te global error O ) as opposed to te simple Euler s O)), or equivalently, te local truncation error O 3 ). Below we will determine te value of A. You will be asked to compute B along similar lines in one of te omework problems. To determine te coeficient A in te Modified Euler metod 1.18), let us rewrite tat equation wile Taylor-expanding its r..s. using Eq. 0.6) wit x = and y = A: Y i+1 = Y i + fx i, Y i ) + fxi, Y i ) + [f x x i, Y i ) + A)f y x i, Y i )] + O ) ) = Y i + fx i, Y i ) + f xx i, Y i ) + Af y x i, Y i )) + O 3 ). 1.0) Equation 1.0) yields te Taylor expansion of te computed solution Y i+1. Let us compare it wit te Taylor expansion of te exact solution yx i+1 ). To simplify te notations, we will denote y i = y x i ), etc. Ten, using Eq. 1.7): y i+1 = y i + y i + y i + O 3 ) = y i + fx i, y i ) + f xx i, y i ) + fx i, y i )f y x i, y i )) + O 3 ). 1.1) Upon comparing te last lines of Eqs. 1.0) and 1.1), we see tat in order for metod 1.18) to ave te local truncation error of O 3 ), one sould take A = fx i, Y i ).

7 MATH 337, by T. Lakoba, University of Vermont 1 Tus, te Modified Euler metod can be programmed into a computer code as follows: Y 0 = y 0, Ȳ i+1 = Y i + fx i, Y i ), Y i+1 = Y i + fxi, Y i ) + fx i+1, Ȳi+1) ) 1.). Remark: An alternative way to code in te last line of te above equation is Y i+1 = 1 Yi + Ȳi+1 + fx i+1, Ȳi+1) ). 1.3) Tis way is more efficient, because it requires only one evaluation of function f, wic is usually te most time-consuming operation, wile te last line of 1.) requires two function evaluations. In a omework problem, you will sow tat in Eq. 1.19), B = 1fx i, Y i ). Ten te Midpoint metod can be programmed as follows: Y 0 = y 0, Y i+ 1 = Y i + fx i, Y i ), Y i+1 = Y i + f x i + ), Y i ) Bot te Modified Euler and te Midpoint metods ave te local truncation error of O 3 ) and te discretization and global errors of O ). Tus, tese are te nd-order metods. Te derivation of te local truncation error for te Modified Euler metod is given in te Appendix to tis section. Tis derivation will be needed for solving some of te omework problems. Remark about notations: Different books use different names for te metods wic we ave called te Modified Euler and Midpoint metods. 1.5 An alternative way to improve te accuracy of a finite-difference metod: Ricardson metod / Romberg extrapolation We ave sown tat te global error of te simple Euler metod is O), wic means tat Y i = y i + O) = y i + a + b +...) = y i + a + O ) 1.5) were a, b, etc. are some constant coefficients tat depend on te function f and its derivatives as well as on te values of x), but not on. Te superscript in Yi means tat tis particular numerical solution as been computed wit te step size. We can now alve te step size and re-compute Y / i, wic will satisfy Y / i = y i + a ) ) + b +... = y i + a ) + O ). 1.6) Let us clarify tat Y / i is not te numerical solution at x i + /) but rater te numerical solution computed from x 0 up to x i wit te step size /).

8 MATH 337, by T. Lakoba, University of Vermont 13 Equations 1.5) and 1.6) form a system of linear equations for te unknowns a and y i. Solving tis system, we find y i = Y / i Yi + O ). 1.7) Tus, a better approximation to te exact solution tan eiter Y i or Y / i is Y improved Y / i i = Yi. Te above metod of improving accuracy of te computed solution is called eiter te Romberg extrapolation or Ricardson metod. It works for any finite-difference sceme, not just for te simple Euler. However, it is not computationally efficient. For example, to compute Y improved as per Eq. 1.7), one requires one function evaluation to compute Yi+1 from Yi and two function evaluations to compute Y / / i+1 from Yi since we need to use two steps of size / eac). Tus, te total number of function evaluations to move from point x i to point x i+1 is tree, compared wit two required for eiter te Modified Euler or Midpoint metods. 1.6 Appendix: Derivation of te local truncation error of te Modified Euler metod Te idea of tis derivation is te same as in Section 1., were we derived an estimate for te local truncation error of te simple Euler metod. Te details of te present derivation, owever, are more involved. In particular, we will use te following formula, obtained similarly to 1.7): y x) = d3 yx) dx 3 use te ODE = d fx, y) dx use 1.7) = f x + f y f) x dx dx + f x + f y f) y dy dx use te Product rule = f xx + f x f y + ff xy + ff y ) + f f yy. 1.8) Let us recall tat in deriving te local truncation error at point x i+1, one always assumes tat te exact solution y i and te computed solution Y i at te previous step i.e. at point x i ) are equal: y i = Y i. Also, for brevity of notations, we will write f witout arguments to mean eiter fx i, y i ) or fx i, Y i ): f fx i, y i ) = fx i, Y i ). By te definition, given in Section 1., te local truncation error of te Modified Euler metod is computed as follows: ɛ ME i+1 = y i+1 Yi+1 ME, 1.9) were y i+1 and Y i+1 are te exact and computed solutions at point x i+1, respectively assuming tat y i = Y i ). We first find y i+1 using ODE 1.1): y i+1 = yx i + ) = y i + y i + y i y i + O 4 ) use 1.7) and 1.8) = y i + f + f x + ff y ) fxx + f x f y + ff xy + ff y ) + f f yy ) + O 4 ).1.30)

9 MATH 337, by T. Lakoba, University of Vermont 14 We now find Y ME i+1 from Eq. 1.): Y ME i+1 = Y i + f + fx i +, Y i + f) ) for last term, use 0.6) wit x= and y=f = Y i + { f + f + [f x + ff y ] + 1 } )! [ f xx + f f xy + f) f yy ] + O 3 ) = Y i + f + f x + ff y ) f xx + ff xy + f f yy ) + O 4 ). 1.31) Finally, subtracting 1.31) from 1.30), one obtains: [ 1 ɛ ME i+1 = ) f xx + ff xy + f f yy ) + 16 ] 4 f x + ff y )f y + O 4 ) [ = f xx + ff xy + f f yy ) + 1 ] 6 f x + ff y )f y + O 4 ). 1.3) For example, let fx, y) = ay, were a = const. Ten f x = f xx = f xy = 0, f y = a, and f yy = 0, so tat from 1.3) te local truncation error of te Modified Euler metod, applied to te ODE y = ay, is found to be ɛ ME i+1 = 3 6 a3 y + O 4 ). 1.7 Questions for self-assessment 1. Wat does te notation O k ) mean?. Wat are te meanings of te local truncation error, discretization error, and global error? 3. Give an example wen te triangle inequality 1.5) olds wit te < sign. 4. Be able to explain all steps made in te derivations in Eqs. 1.15) and 1.16). 5. Wy are te Modified Euler and Midpoint metods called nd-order metods? 6. Obtain 1.7) from 1.5) and 1.6). 7. Explain wy te properly programmed Modified Euler metod requires exactly two evaluations of f per step. 8. Wy may one prefer te Modified Euler metod over te Romberg extrapolation based on te simple Euler metod?