Numerical Analysis of Differential Equations Numerical Solution of Parabolic Equations

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1 Numerical Analysis of Differential Equations Numerical Solution of Parabolic Equations 6 Numerical Solution of Parabolic Equations TU Bergakademie Freiberg, SS 2012

2 Numerical Analysis of Differential Equations The One-Dimensional Model Problem We consider the following initial boundary value problem (IBVP) modelling heat flow in a thin rod, i.e., in one space dimension: u t = κu xx, x (0, 1), t > 0, (6.1a) u(0, t) = g 0 (t), t > 0, (6.1b) u(1, t) = g 1 (t), t > 0, (6.1c) u(x, 0) = u 0 (x), x [0, 1] (6.1d) with given (constant) heat conductivity κ (which we set to one in the following) as well as (possibly time-dependent) Dirichlet boundary values g 0, g 1 and initial data u The One-Dimensional Model Problem TU Bergakademie Freiberg, SS 2012

3 Numerical Analysis of Differential Equations 217 Steady-state version: u xx = 0, u(0) = g 0, u(1) = g 1. Analogous IBVP in 2D and 3D: u t = u + initial and boundary data. Related: linear ordinary differential equation u t = Au, u : t u(t) R n. Here linear differential operator xx in place of matrix A. 6.1 The One-Dimensional Model Problem TU Bergakademie Freiberg, SS 2012

4 Numerical Analysis of Differential Equations 218 Series solution by separation of variables: in special cases an analytic representation of the (exact) solution may be constructed using the technique of separation of variables. This is helpful for checking numerical approximations and provides important insight into the structure of the solution. Inserting the special trial solution u(x, t) = f(x)g(t) into the PDE u t = u xx results in fg = f g, i.e. For each value of k we obtain a solution g g = f f = const. =: k2. u k (x, t) = e k2t sin(kx) of u t = u xx. The boundary condition u(0, t) = u(1, t) = 0 constrains k to the discrete values k = k m := mπ, m N. 6.1 The One-Dimensional Model Problem TU Bergakademie Freiberg, SS 2012

5 Numerical Analysis of Differential Equations 219 Due to the linearity and homogeneity of u t = u xx any linear combination of these solutions is also a solution. If we succeed in finding coefficients a m in such a way that u 0 (x) = a m sin(mπx), then the series u(x, t) := solves the complete IBVP (6.1). m=1 m=1 a m e m2 π 2t sin(mπx) Since the functions {sin(mπx)} m=1 form a complete orthogonal system of the function space L 2 (0, 1), this is possible for all u 0 L 2 (0, 1). The coefficients are given by a m = u 0 (x) sin(mπx) dx. 6.1 The One-Dimensional Model Problem TU Bergakademie Freiberg, SS 2012

6 Numerical Analysis of Differential Equations 220 Example: u 0 (x) = 1 2 x 1 2. Here the coefficients are a m = 8 mπ m 2 sin π2 2, m N N=1 N=3 N=5 N=19 u 0 (x) x Partial sums N m=1 a m sin(mπx) of the Fourier series of u The One-Dimensional Model Problem TU Bergakademie Freiberg, SS 2012

7 Numerical Analysis of Differential Equations t=0 t=1/ m Decay of first 4 nonzero Fourier coefficients of u x 0 1 Fourier series solution of IBVP (6.1) with κ = 1 and u 0 (x) = 1 2 x 1 2 in domain (x, t) [0, 1] [0, 1] t The One-Dimensional Model Problem TU Bergakademie Freiberg, SS 2012

8 Numerical Analysis of Differential Equations 222 We first discretize the IBVP (6.1) in the spatial variable x only, leaving time t continuous. To this end we proceed as in the elliptic case and introduce the grid points 0 = x 0 < x 1 < < x J < x J+1 = 1 using a fixed grid spacing x = 1/(J + 1) and approximate u xx x=xj [A x u] j, j = 1, 2,..., J, with A x = 1 tridiag(1, 2, 1). (6.2) x2 If u = u(t) denotes the vector with components u j (t) u(x j, t), t > 0, j = 1, 2,..., J, 6.1 The One-Dimensional Model Problem TU Bergakademie Freiberg, SS 2012

9 Numerical Analysis of Differential Equations 223 then (6.1) is transformed into the semi-discrete system of ODEs u (t) = A x u(t) + g(t), u(0) = u 0, (6.3a) (6.3b) with [u 0 ] j = u 0 (x j ), j = 1, 2,..., J as well as g(t) = 1/( x 2 )[g 0 (t), 0,..., 0, g 1 (t)] R J. We can now solve (6.3) with known numerical methods for solving ODEs. Introducing the fixed time step t > 0, we set U n j [u(t n )] j u(x j, t n ), t n = n t. The approximation of the solution of a time-dependent PDE as a system of ODEs along the lines {(x j, t) : t > 0} is known as the method of lines. 6.1 The One-Dimensional Model Problem TU Bergakademie Freiberg, SS 2012

10 Numerical Analysis of Differential Equations 224 Applying the explicit Euler method to (6.3) (setting g 0 (t) = g 1 (t) = 0 for now) leads to U n+1 j = Uj n + t ( U n x 2 j 1 2Uj n + Uj+1 n ), 1 j J, n = 0, 1, This corresponds to the finite difference approximation Uj n } t {{ } u t U n+1 j = U n j 1 2U n j + U n j+1 x 2 }{{} u xx (6.4) of the differential equation (6.1a). In matrix notation: U n+1 = (I + t A x )U n, n = 0, 1, 2,..., with U n = [U1 n, U2 n,..., UJ n]. 6.1 The One-Dimensional Model Problem TU Bergakademie Freiberg, SS 2012

11 Numerical Analysis of Differential Equations 225 We define the local discretisation error of the difference scheme (6.4) to be the residual obtained on inserting the exact solution into the difference scheme: d(x, t) := u(x, t + t) u(x, t) t u(x x, t) 2u(x, t) + u(x + x, t) x 2. Using Taylor expansions in (x, t) one easily obtains: ( ) t d ee (x, t) = 2 x2 u xxxx + O( t 2 ) + O( x 4 ). (6.5) 12 Terminology: the explicit Euler method for solving the heat equation is consistent of first order in time and of second order in space. 6.1 The One-Dimensional Model Problem TU Bergakademie Freiberg, SS 2012

12 Numerical Analysis of Differential Equations 226 Besides the asymptotic statement (6.5) we also require upper bounds for d. Truncating the Taylor expansions with a remainder term, we obtain in time u(x, t + t) = (u + t u t ) (x,t) + t2 2 u tt(x, τ), τ (t, t + t). Proceeding analogously in x yields d ee (x, t) = t 2 u tt(x, τ) x2 12 u xxxx(ξ, t), ξ (x x, x + x), and we obtain, setting µ := t x 2, d ee (x, t) t 2 M tt x2 12 M xxxx = t 2 assuming u tt M tt und u xxxx M xxxx on [0, 1] [0, T ]. ( M tt + 1 ) 6µ M xxxx, (6.6) 6.1 The One-Dimensional Model Problem TU Bergakademie Freiberg, SS 2012

13 Numerical Analysis of Differential Equations 227 Applying the implicit Euler method to (6.3) one obtains, in place of (6.4), the implicit difference scheme U n+1 j t U n j = U n+1 j 1 n+1 2Uj + U n+1 j+1 x 2. (6.7) The calculation of u n+1 from u n is seen to require the solution of a linear system of equations with coefficient matrix I ta x. Here Taylor expansion results in a local discretization error of ( ) t d ie (x, t) = 2 + x2 u xxxx + O( t 2 ) + O( x 4 ) The One-Dimensional Model Problem TU Bergakademie Freiberg, SS 2012

14 Numerical Analysis of Differential Equations 228 Applying instead the trapezoidal rule, which for an ODE y (t) = f (t, y(t)) is given by y n+1 = y n + t 2 [ f (tn, y n ) + f (t n+1, y n+1 ) ] yields another implicit scheme U n+1 j Uj n = 1 ( U n j 1 2Uj n + U j+1 n t 2 x 2 + U n+1 j 1 n+1 2Uj + U n+1 ) j+1 x 2, which in this context is known as the Crank-Nicolson scheme a. This method is also implicit, requiring in each time step the solution of a linear system of equations with the coefficient matrix I t 2 A x. For Crank-Nicolson (CN) there holds d CN (x, t) = O( t 2 ) + O( x 2 ). a J. CRANK AND P. NICOLSON (1947) 6.1 The One-Dimensional Model Problem TU Bergakademie Freiberg, SS 2012

15 Numerical Analysis of Differential Equations Convergence All three methods considered so far are consistent, i.e., at all points (x, t) of the domain we have d(x, t) 0 as x 0 and t 0. To analyze their convergence, we proceed as in the case of numerical methods for ODEs and consider a finite time interval t [0, T ], T > 0 as well as a sequence of grids with grid spacings x 0, t 0 and determine whether at every fixed grid point (x j, t n ) also the global error u(x j, t n ) Uj n tends to zero uniformly. A sequence of grid spacings {( x) k, ( t) k } can approach the point (0, 0) in different ways. The following figure shows different refinement curves in the ( x, t)-plane. We will see that the explicit Euler method converges only if the refinement satisfies µ := t x Convergence TU Bergakademie Freiberg, SS 2012

16 Numerical Analysis of Differential Equations t = x 2 /4 t = x 2 /2 t = x 2 t = x t = x/2 0.3 t x Different refinement curves t, x 0: red: t/ x 2 = const, blue: t/ x = const. 6.2 Convergence TU Bergakademie Freiberg, SS 2012

17 Numerical Analysis of Differential Equations 231 Theorem 6.1 If a sequence of grids satisfies µ k = ( t) k ( x) 2 k 1 2 for all k sufficiently large, and if for the corresponding sequences {j k } and {n k } there holds n k ( t) k t [0, T ], j k ( x) k x [0, 1], then, under the assumption that u xxxx M xxxx uniformly in [0, 1] [0, T ], the approximations generated by the explicit Euler method (6.4) converge to the exact solution u(x, t) as k uniformly on [0, 1] [0, T ]. 6.2 Convergence TU Bergakademie Freiberg, SS 2012

18 Numerical Analysis of Differential Equations expl. Euler dx=1/10 expl. Euler dx=1/20 Crank Nicolson, dx = 1/10 Crank Nicolson, dx = 1/ E n t Error E(t n ) = max j u n j U n j of numerical solution of model problem(6.1) with g 0 = g 1 = 0 and u 0 (x) = x(1 x) obtained with explicit Euler method with t = x 2 /2 and the Crank-Nicolson method with t = x/ Convergence TU Bergakademie Freiberg, SS 2012

19 Numerical Analysis of Differential Equations Stability In the method of lines, an ODE solver is applied to the semidiscrete system (6.3). Recalling the regions of absolute stability of the explicit and implicit Euler and trapezoidal methods, we are led to investigate the behavior of the eigenvalues of the matrix A x in (6.2) as x explicit Euler R( λ) = 1 + λ implicit Euler R( λ) = 1 1 λ trapezoidal R( λ) = 1+ λ/2 1 λ/2 6.3 Stability TU Bergakademie Freiberg, SS 2012

20 Numerical Analysis of Differential Equations 234 The eigenvalues of A x (cf. (5.7)) are given by λ j = 4 jπ x x 2 sin2, j = 1, 2..., J. 2 The eigenvalue of largest magnitude is λ J = 4 ( π ) x 2 sin2 2 (1 x) Therefore, in order to satisfy = 4 x 2 + π2 + O ( x 2). R ( t λ j ) 1 for all j = 1, 2,..., J in the explicit Euler method, it is necessary that t x (6.8) Both implicit Euler and the trapezoidal rule are absolutely stable methods. Therefore, as all eigenvalues of A x lie in the left half plane, the requirement of absolute stability places no constraints on the time step t. Since the CN scheme (trapezoidal rule) is consisten of order 2 in both x and t, this suggests a time step t x. 6.3 Stability TU Bergakademie Freiberg, SS 2012

21 Numerical Analysis of Differential Equations 235 Absolute stability describes the behavior of a numerical approximation u n at time t n of the solution of an ODE as n. The appropriate stability concept for the convergence analysis of numerical methods for solving IVPs for PDEs was developed by Lax a and Richtmyer b. All three methods considered so far can be written in the form c U n+1 = B( t)u n + g n ( t), n = 0, 1,... (6.9) with different matrices B( t) given in each case by B ee ( t) = I + t A x, (6.10a) B ie ( t) = (I t A x ) 1, ( B CN ( t) = I t ) 1 ( 2 A x I + t ) 2 A x a PETER D. LAX ( 1906) b ROBERT DAVIS RICHTMYER ( ) c We assume that t is chosen as a fixed given function of x. (6.10b). (6.10c) 6.3 Stability TU Bergakademie Freiberg, SS 2012

22 Numerical Analysis of Differential Equations 236 A linear finite difference scheme of the form (6.9) is called Lax-Richtmyer stable, if for any fixed stopping time T there exists a constant K T > 0 such that B( t) n K T, t > 0 and n N, n t T. Theorem 6.2 (Lax-Richtmyer equivalence theorem) A consistent linear finite difference scheme (6.9) is convergent if, and only if, it is Lax-Richtmyer stable. Examples: For the explicit Euler method applied to (6.1) the resulting matrix (6.10a) is normal, implying B( t) 2 = ρ(b( t)). If condition (6.8) is satisfied, we have B( t) 2 1, implying that the explicit Euler method is Lax-Richtmyer stable and therefore convergent. The implicit Euler and Crank-Nicolson schemes are also Lax-Richtmyer stable but without constraints on the mesh ratio or time step. 6.3 Stability TU Bergakademie Freiberg, SS 2012

23 Numerical Analysis of Differential Equations 237 Remark 6.3 In all three schemes considered so far we were able to show the stronger statement B( t) 1, which is sometimes called strong stability. However, strong stability is not necessary for Lax-Richtmyer stability. More precisely, for Lax-Richtmyer stability it is necessary that there exist a constant α such that B( t) 1 + α t (6.11) for all sufficiently small time steps t, which is evident from the inequality B( t) n (1 + α t) n e αt, n t T. 6.3 Stability TU Bergakademie Freiberg, SS 2012

24 Numerical Analysis of Differential Equations Von-Neumann Analysis A formal procedure based on Fourier analysis for determining stability bounds was introduced by von Neumann a. In general, von Neumann analysis only yields necessary conditions, which are also sufficient only in special cases, among these linear PDEs with constant coefficients, unbounded spatial domain (Cauchy problem) or periodic boundary conditions, uniform grid. Fundamental fact: in the continuous case (x R) the functions e iξx, ξ R, are eigenfunctions of the differential operator x, i.e., x e iξx = iξ e iξx, and thus are also eigenfunctions of any linear differential operator with constant coefficients. a JOHN VON NEUMANN ( ) 6.4 Von-Neumann Analysis TU Bergakademie Freiberg, SS 2012

25 Numerical Analysis of Differential Equations 239 On the unbounded spatial grid {x j = j x} j Z the grid function v j := e iξx j = e iξj x, j Z, is an eigenfunction of any difference operator with constant coefficients. Example: For the central difference operator δ 0 defined as (δ 0 v) j := (v j+1 v j 1 )/(2 x) there holds δ 0 v j := (δ 0 v) j = eiξ(j+1) x e iξ(j 1) x 2 x = i x sin(ξ x) v j The power series expansion = eiξ x e iξ x 2 x e iξj x i x sin(ξ x) = i x (ξ x (ξ x)3 6 + O ( (ξ x) 5)) = iξ + O( x 2 ξ 3 ) shows that the eigenvalue of δ 0 approximates the corresponding eigenvalue of x. 6.4 Von-Neumann Analysis TU Bergakademie Freiberg, SS 2012

26 Numerical Analysis of Differential Equations 240 Any grid function on {x j = j x : j Z} can be represented as the superposition v j = 1 2π π/ x π/ x v(ξ)e iξj x dξ, j Z of grid waves e iξj x, ξ [ π, π ] with Fourier coefficients v(ξ) given by x x v(ξ) = x j= v j e iξj x, ξ [ π x, π ]. x If the magnitude of grid functions and their Fourier coefficients are measured by v = ( x j= v j 2 ) 1/2 then Parseval s identity states that and v = ( π/ x π/ x v(ξ) 2 dξ) 1/2, v = 2π v. (6.12) 6.4 Von-Neumann Analysis TU Bergakademie Freiberg, SS 2012

27 Numerical Analysis of Differential Equations 241 The bound B 1 + α t requires verifying Bu (1 + α t) u u. According to (6.12), an equivalent condition is Bu (1 + α t) û û, (6.13) which is easier to show because the Fourier coefficients are decoupled. For a one-step method one obtains û n+1 (ξ) = g(ξ) û n (ξ) with a so-called amplification factor g(ξ). If we are able to show that g(ξ) 1 + α t. with a constant α which is independent of ξ, then this implies (6.13). 6.4 Von-Neumann Analysis TU Bergakademie Freiberg, SS 2012

28 Numerical Analysis of Differential Equations 242 Examples: (a) For the explicit Euler scheme applied to (6.3) we obtain the amplification factor g(ξ) = 1 4 t ξ x sin2 x2 2. (b) For Crank-Nicolson, g(ξ) = 1 + z(ξ) t ( ), z(ξ) = 1 z(ξ) x cos(ξ x) 0 ξ. 6.4 Von-Neumann Analysis TU Bergakademie Freiberg, SS 2012