Finite Volume Discretization of the Heat Equation

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1 Lecture Notes 3 Finite Volume Discretization of te Heat Equation We consider finite volume discretizations of te one-dimensional variable coefficient eat equation, wit Neumann boundary conditions u t x (k(x) x u) = S(t, x), 0 < x <, t > 0, () Te coefficient k(x) is strictly positive. u(0, x) = f(x), 0 < x <, u x (t, 0) = u x (t, ) = 0, t 0. Semi-discrete approximation By semi-discretization we mean discretization only in space, not in time. Tis approac is also called metod of lines. Discretization We discretize space into N equal size grid cells (bins) of size = /N, and define x j = /2 + j, so tat x j is te center of cell j, see figure. Te edges of cell j are ten and x j+/2. x 0 x x N Unknowns In a finite volume metod te unknowns approximate te average of te solution over a grid cell. More precisely, we let q j (t) be te approximation q j (t) u j (t) := u(t, x)dx. Note te contrast wit finite difference metods, were pointwise values are approximated, and finite element metods, were basis function coefficients are approximated. (0)

2 Exact update formula We derive an exact update formula for u j (t), te exact local averages. Integrating () over cell j and dividing by we get u t (t, x)dx = Upon defining te flux and te local average of te source we get te exact update formula du j (t) x (k(x) x u)dx + S(t, x)dx = k(x j+/2)u x (t, x j+/2 ) k( )u x (t, ) = F j+/2(t) F j /2 (t) F j (t) = F (t, x j ) = k(x j )u x (t, x j ), S j (t) = + S j (t). S(t, x)dx, Te fluxes F j+/2 and F j /2 ten represents ow muc eat flows out troug te left and rigt boundary of te cell. Note, tis is an instance te conservation law in integral form, d V udv + S F nds = V SdV, were we ave picked V as te interval [, x j+/2 ], and scaled by V =. Approximation of te flux + S(t, x)dx. To use te exact update formula as te basis for a numerical sceme we must approximate te fluxes F j±/2. Since te value in te midpoint of te cell is a second order approximation of te average, we ave for smoot u, F j /2 (t) = k( )u x (t, ) = k( ) u(t, x j) u(t, x j ) = k( ) u j(t) u j (t) + O( 2 ). + O( 2 ) We terefore use F j /2 (t) F j /2 (t) = k( ) q j(t) q j (t). as approximation. Tis leads to te numerical sceme for inner points j N 2, dq j (t) F j+/2 (t) F j /2 (t) = + S j (t) (2) = k(x j+/2)(q j+ (t) q j (t)) k( )(q j (t) q j (t)) 2 + S j (t). 2 (0)

3 Hence, wit k j := k(x j ), dq j = k j+/2q j+ (k j+/2 + k j /2 )q j + k j /2 q j 2 + S j, (3) for j =,..., N 2. Tis is a second order approximation. Boundary conditions To complete te sceme (3) we need update formulae also for te boundary points j = 0 and j = N. Tese must be derived by taking te boundary conditions into account. We introduce te gost cells j = and j = N wic are located just outside te domain. Te boundary conditions are used to fill tese cells wit values q and q N, based on te values q j in te interior cells. Te same update formula (3) as before can ten be used also for j = 0 and j = N. Let us consider our boundary condition u x = 0 at x = 0. (We can also tink of tis as a "no flux" condition, F = 0.) We formally extend te definition of te solution u for x < 0, i.e. outside te domain, and, as before, approximate 0 = u x (t, 0) = u(t, x 0) u(t, x ) + O( 2 ) = u 0(t) u (t) + O( 2 ) u (t) = u 0 (t) + O( 3 ). Replacing u j by our approximation q j and dropping te O( 2 ) term we get an expression for q in terms of q 0 as te boundary rule q (t) = q 0 (t). We now insert tis into te update formula (3) for j = 0, dq 0 = k /2q (k /2 + k /2 )q 0 + k /2 q 2 + S 0 = k /2 q q S 0. (4) In exactly te same way we obtain for j = N tat q N (t) = q N (t) and terefore dq N = k N 3/2 q N 2 q N 2 + S N. (5) Remark Diriclet boundary conditions u(t, 0) = u(t, ) = 0 can be approximated to second order in two ways. First, one can use a sifted grid, x j = j so tat x 0 and x N, te centers of cells 0 and N, are precisely on te boundary. Ten one does not need gost cells; one just sets q 0 = q N = 0. Note tat te number of unknows are now only N, so A R (N ) (N ) etc. Second, one can take te average of two cells to approximate te value in between, 0 = u(t, 0) = u(t, x 0) + u(t, x ) 2 leading to te approximations + O( 2 ) = u 0(t) + u (t) 2 q = q 0, q N = q N. + O( 2 ), 3 (0)

4 Matrix form We put all te formulae (3), (4), (5) togeter and write tem in matrix form. Introduce q = and k /2 k /2 A = 2 q 0. q N, S = S 0. S N k /2 (k /2 + k 3/2 ) k 3/ Ten we get te linear ODE system, q, S R k N 5/2 (k N 5/2 + k N 3/2 ) k N 3/2 k N 3/2 k N 3/2 dq(t) R N N. (6) = Aq(t) + S(t). (7) Hence, in tis semi-discretization te time-dependent PDE as been approximated by a system of ODEs, were te matrix A is a discrete approximation of te second order differential operator x k(x) x, including its boundary conditions.. Brief outline of extensions to 2D Te same strategy can be used in 2D. We give a cursory description of te main steps ere. To simplify tings we consider te constant coefficient problem, u t u = S(t, x, y), 0 < x <, 0 < y <, t > 0, u(0, x, y) = f(x, y), 0 < x <, 0 < y <, u x (t, 0, y) = u x (t, 2π, y) = u y (t, x, 0) = u y (t, x, 2π) = 0, 0 < x <, 0 < y <, t 0. Discretization We discretize te domain [0, ] 2 into N N equal size grid cells of size, were = /N. We define x j = /2 + j and y k = /2 + k and denote te cell wit center (x j, y k ) by I jk. 4 (0)

5 y N I jk y y 0 x 0 x x N Unknowns Te unknowns are now q jk (t), wic are approximations of te average of te solution over te grid cell I jk, q jk (t) u jk (t) := 2 I jk u(t, x, y)dxdy. Exact update formula Te update formula is again an instance of te conservation law in integral form were we pick te volume V as I jk and scale by I jk = 2, d 2 udxdy + F nds = I jk 2 I jk 2 Sdxdy, I jk were F = u. Upon defining S jk (t) := 2 I jk S(t, x, y)dxdy, tis can be written as du jk (t) = 2 I jk F nds + Sjk (t). Let F = (f, f 2 ) and define te average flux troug eac side of te cell, F j,k±/2 (t) := we get te exact formula f 2 (t, x, y k±/2 )dx, F j±/2,k (t) := yk+/2 y k /2 f (t, x j±/2, y)dy, du jk = (F j+/2,k F j /2,k + F j,k+/2 F j,k /2 ) + S jk. (8) 5 (0)

6 Again, te fluxes F j,k±/2 and F j±/2,k represent te eat flux (upto sign) out troug te four sides of te cell. Approximation of te flux We use te same type of approximation as in D for F = u. We get for instance F j,k /2 = = 2 u y (t, x, y k /2 )dx = u(t, x, y k )dx + 2 = 3 I jk u(t, x, y)dxdy + 3 = u jk + u j,k + O( 2 ). u(t, x, y k ) u(t, x, y k ) u(t, x, y k )dx + O( 2 ) I j,k u(t, x, y)dxdy + O( 2 ) dx + O( 2 ) Replacing u jk by q jk, dropping O( 2 ) and inserting in te exact formula (8) we get te five-point formula, dq jk = 2 (q j+,k + q j,k + q j,k+ + q j,k 4q jk ) + S jk. Boundary conditions is done as in D. Te matrix form is more complicated and te subject of Homework. 2 Properties of te semi-discrete approximation 2. Conservation Wen S = 0 in () we ave seen tat te solution as te conservation property 0 u(t, x)dx = constant = 0 f(x)dx. In fact tis olds for all conservation laws wit "no-flux" boundary conditions F = 0, wic is easily seen from te integral form of te PDE. An analogue of tis olds also for te semi-discrete approximation. More precisely, let us define Ten Q(t) = N q j (t). Q(t) = constant. (9) Moreover, if we start te approximation wit exact values, q j (0) = u j (0), ten it follows tat Q(t) is exactly te integral of te solution u for all time, Q(t) = constant = Q(0) = N u j (0) = N We prove te exact discrete conservation (9) in two ways. 6 (0) u(0, x)dx = 0 u(0, x)dx = 0 u(t, x)dx.

7 Recall from (2) tat wen S = 0, Ten N dq = dq j (t) N dq j = F j+/2 (t) F j /2 (t) =. (0) F j /2 (t) F j+/2 = F /2 F N /2 = 0, were we used te boundary conditions (4) and (5) wic implies tat F /2 = k /2 q 0 q = 0, FN /2 = k N /2 q N q N Note tat tis discrete conservation property is true for any discretization of te type (0) if F /2 = F N /2, regardless of ow te fluxes F j±/2 are computed. Recall from (7) tat wen S = 0, Moreover, dq(t) = Aq(t). Q(t) = T q, =. RN. Ten dq(t) = T dq(t) = T Aq(t) = 0, since T Aq(t) = (A) T q(t) by te symmetry of A and A are te row sums of A wic are zero, by (6). 2.2 Maximum principle We ave seen before tat in te continuous case tat wen S = 0 in (), te maximum value of te solution u(t, x) in [0, T ] [0, ] is eiter attained on te boundary x {0, } or for te initial data at t = 0 (regardless of boundary conditions). Te corresponding result in te semi-discrete case says tat q = max sup q j (t) j 0,...,N 0 t<t is attained eiter for j = 0, j = N or for te initial data t = 0. Tis is easily sown by contradiction. Suppose te maximum is attained at t = t > 0 and tat it is strictly larger tan q 0 (t ) and q N (t ). Ten, tere must be an interior index j [, N 2] suc tat Terefore dq j (t ) q j (t ) < q j (t ), q j (t ) q j +(t ). = k j +/2 (q j + q j ) }{{} 0 k j /2 (q j q j ) < 0. }{{} >0 Hence, tere is an ε suc tat q j (t) > q j (t ) for all t (t ε, t ), wic contradicts te assumption tat q j (t ) is a maximum. Furtermore, as in te continuous case local spatial maximum (minimum) of q j in te interior cannot increase (decrease) in time. = 0. 7 (0)

8 Forward Euler Im Backward Euler Im D D Re Re Figure. Stability regions D for te Forward and Backward Euler metods. 3 Fully discrete approximation Te semi-discrete approximation leads to a system of ODEs. dq(t) = Aq(t) + S(t). () Tis can be solved by standard numerical metods for ODEs wit a time step t, e.g. Forward Euler metod u n+ = u n + tf(t n, u n ), t n = n t. Applied to () tis would give te fully discrete sceme for (), q n+ = q n + t[aq n + S(t n )], q n q(t n ). As for any ODE metod we must verify its absolute stability: tat for our coice of step size t te tλ k D, k, were D is te stability region of te ODE solver and {λ k } are te eigenvalues of A. For parabolic problems te real part of te eigenvalues are negative, and te size of tem in general grow as / 2. Tis is a major difficulty. It means tat wen te stability region D is bounded, as in explicit metods, we get a time step restriction of te type t C 2. Tis is a severe restriction, wic is seldom warranted from an accuracy point of view. It leads to unnecessarily expensive metods. Example In te constant coefficient case k(x) te eigenvalues of A are precisely λ k = 4 ( ) kπ 2 sin2, k = 0,..., N. 2 Tis gives te stability condition t 2 /2 for Forward Euler. 8 (0)

9 Te underlying reason for te severe time-step restriction is te fact tat parabolic problems include processes on all time-scales: ig frequencies decay fast, low frequencies slowly. Teir semi-discretization ave time-scales spread out all over te interval [ / 2, 0], wic means tat te ODEs are stiff. Te consequence is tat implicit metods sould be used for parabolic problems. Implicit metods typically ave unbounded stability domains D and ave no stability restriction on te time-step tey are unconditionally stable. Of course, implicit metods are more expensive per time-step tan explicit metods, since a system of equations must be solved, but tis is outweiged by te fact tat muc longer time-steps can be taken. Moreover, w en te coefficients do not vary wit time, matrices etc. can be constructed once, and re-factored only as canges of time-step make it necessary. Te "θ-metod" is a class of ODE metods defined as u n+ = u n + t [ θf(t n+, u n+ ) + ( θ)f(t n, u n ) ], 0 θ. Tis includes some common metods: Applied to () we ave or θ = 0 Forward Euler (explicit, st order), θ = /2 Crank Nicolson (implicit, 2nd order), θ = Backward Euler (implicit, st order), q n+ = q n + ta[θq n+ + ( θ)q n ] + t [θs(t n+ ) + ( θ)s(t n )], (2) }{{} S n θ ( θ ta)q n+ = ( + ( θ) ta)q n + ts n θ. For te constant coefficient problem k one can sow te time-step restriction t 2 { 2( 2θ), θ < /2,, /2 θ, (unconditionally stable). Remark 2 Te Crank-Nicolson sceme is second order accurate but gives slowly decaying oscillations for large eigenvalues. It is unsuitable for parabolic problems wit rapidly decaying transients. Te θ = sceme damps all components, and sould be used in te initial steps. Remark 3 Te most used family of time-stepping scemes for parabolic problems are te Backward Differentiation Formulas (BDF), of order troug 5 wic are A(α)-stable. Tey are multistep metods generalizing Backward Euler to iger order. For instance, te second order BDF metod is u n+ = 4 3 un 3 un tf(t n+, u n+ ). BDF metods are also known as Gear s metods and available in Matlab as ODE5S. 3. Fully discrete conservation For te θ-metod we also ave discrete conservation wen S 0. Let Q n N q n j = T q n. 9 (0)

10 Ten upon multiplying by from te left in (2) we get Q n+ = T q n+ = T q n + t T A[θq n+ +( θ)q n ] = Q n + t(a) T [θq n+ +( θ)q n ] = Q n, since as before A = 0. In particular, if initial data is exact, q 0 j = u j(0), ten Q n = 0 u(t n, x)dx, for all n 0. Te same is true for te second order BDF metod if te initialization of te first step Q is conservative so tat Q = Q 0. Ten q n+ = 4 3 qn 3 qn taqn+ implies Q n+ = T q n+ = 4 3 T q n 3 T q n T taq n+ = 4 3 Qn 3 Qn. Wit te stipulated initial data tis difference equation as te solution Q n = Q 0 for all n 0. 4 Acknowledgement Part of tese notes are based on earlier notes by Prof. Jesper Oppelstrup. 0 (0)

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