Code: 101MAT4 101MT4B. Today s topics Finite-difference method in 2D. Poisson equation Wave equation

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1 Code: MAT MTB Today s topics Finite-difference method in D Poisson equation Wave equation

2 Finite-difference method for elliptic PDEs in D Recall that u is the short version of u x + u y Dirichlet BVP: u = f in Ω = (a, b ) (a, b ), u = g on Γ. Discretization in the x -direction: step size h, coordinates x i = a + ih, i =,,...,M. Discretization in the x -direction: step size h, coordinates x j = a + jh, j =,,...,N. Mesh nodes: P i,j (x i, x j ). Notation: u i,j = u(p i,j ). If u is sufficiently smooth, then its partial derivatives can be well approximated by finite differences:

3 u (P i,j ) = u i+,j u i,j +O(h x h ), u x (P i,j ) = u i,j u i,j + u i+,j h +O(h ), u (P i,j ) = u i,j+ u i,j +O(h x h ), u x (P i,j ) = u i,j u i,j + u i,j+ h +O(h ). Using f i,j f(p i,j ) and U i,j instead of u i,j, we can write finite-difference equations at inner nodes P i,j. Instead of u(p i,j ) = f(p i,j ), we write

4 U i,j U i,j + U i+,j h U i,j U i,j + U i,j+ h = f i,j, () where i =,,...,M, j =,,...,N. The values U,j, U M,j, where j =,,...,N, and the values U i,, U i,n, where i =,,...,M, are given by the boundary condition u = g on Γ. Applying () at each non-boundary mesh node, we obtain (M )(N ) linear algebraic equations with the unknowns U,, U,,..., U,N, U,, U,,...,U,N,..., U M,, U N,,...,U M,N. The unknowns and f i,j are reordered to form a column vector û and f, respectively. The coefficients of the equations form a sparse matrix A. We solve Aû = f by the Gaussian elimination method or by an iterative method. It can be shown that A is s.p.d. If h h = h, then the equations are simpler.

5 Special case: u =, h = h. The equation U i,j U i,j + U i+,j h U i,j U i,j + U i,j+ h = f i,j, where h = h and f i,j =, becomes (U i,j U i,j + U i+,j ) (U i,j U i,j + U i,j+ ) =, that is, U i,j = U i,j + U i+,j + U i,j + U i,j+ ; the mean. A special iterative method Liebmann iteration: at the beginning, U i,j given by boundary conditions or chosen (as, say, zero); the means are calculated in a loop; if a convergence criterion is met, the calculation ends. Gauss-Seidel method. The approximate solution converges to the exact one if h, h.

6 Liebmann iteration: Example u = v Ω = (, ) (,, 5), u = x y + na Γ Ω, h = / in x- and y-direction. Liebmannova iterace. Presne reseni: x y + Liebmannova iterace. Presne reseni: x y osa y osa y

7 Liebmannova iterace. Presne reseni: x y + Liebmannova iterace. Presne reseni: x y osa y osa y Liebmannova iterace. Presne reseni: x y + Liebmannova iterace. Presne reseni: x y osa y osa y

8 Liebmannova iterace. Presne reseni: x y + Liebmannova iterace. Presne reseni: x y osa y osa y Liebmannova iterace. Presne reseni: x y + Liebmannova iterace. Presne reseni: x y osa y osa y

9 Liebmannova iterace. Presne reseni: x y + Liebmannova iterace. Presne reseni: x y osa y osa y Liebmannova iterace. Presne reseni: x y osa y

10 At the end of the.,., 3., and. iteration loop: Liebmannova iterace. Presne reseni: x y + Liebmannova iterace. Presne reseni: x y osa y osa y Liebmannova iterace. Presne reseni: x y + Liebmannova iterace. Presne reseni: x y osa y osa y

11 Liebmannova iterace. Presne reseni: x y osa y Results on a finer mesh; iterations and iterations: Liebmannova iterace. Presne reseni: x y + Liebmannova iterace. Presne reseni: x y osa y.5 osa y

12 Liebmann iteration can solve the Poisson equation u = f Finite-difference equation at P i,j U i,j U i,j + U i+,j h U i,j U i,j + U i,j+ h = f i,j, where h h = h, becomes (U i,j U i,j + U i+,j ) (U i,j U i,j + U i,j+ ) = h f i,j, that is, U i,j = U i,j + U i+,j + U i,j + U i,j+ + h f i,j. Example: u = sin(xy)(y + x ), the BCs are such that the exact solution is u(x, y) = sin(xy) on (, ) (,, 5).

13 Priblizne reseni (presne reseni sin(xy))

14 Priblizne reseni (presne reseni sin(xy))

15 Rozdil mezi presnym a pribliznym resenim v uzlech site x 3 x 3.5 y.5 The difference between the exact and the approximate solution..5 x 3

16 Halving the parameter h the error is reduced by /, more iterations, however. Abs. hodn. rozdilu mezi presnym a pribl. resenim v uzlech site. h=, h h/ h/ h/8 h/ Pocet cyklu Liebmannovych iteraci

17 Wave equation (string vibration) (hyperbolic PDE) u t = u a x v Ω = (, L) (, T), u(x, ) = g (x), < x < L initial displacement u t (x, ) = g (x), < x < L initial velocity u(, t) = u(l, t) =, < t < T boundary condition, a = F ρ, where F is the tension force and ρ is the specific weight of the string related to the length unit.

18 Discretization Mesh nodes: (x i, t k ) = (ih, kτ), i =,,...,M, k =,,...,N, h = L/M, τ = T/N, N and M are natural numbers. u x (x i, t k ) = uk i uk i + ui+ k h +O(h ), u t (x i, t k ) = uk+ i ui k + u k i τ +O(τ ). Finite-difference equation U k+ i Ui k + U k i τ = a Uk i Uk i + Ui+ k h, k =,,...,N, i =,,...,M. Explicit five-point stencil

19 The finite-difference equation gives U k+ i = ( τ a h k =,,...,N, i =,...,M. If τ = h a, we obtain U k+ i ) U ki +τ a ( ) h Ui k + Uk i+ U k i, = U k i + Uk i+ Uk i, k =,,...,N, i =,,...,M (explicit four-point stencil).

20 Boundary condition U k = = Uk M, k =,,...,N. Initial displacement Ui = g (x i ), i =,,...,M. Initial velocity U i = U i +τg (x i ), i =,,...,M. The error of the method: O(τ + h ).

21 Stability of the method by example Struna v case

22 τ =.9 τ kriticke =. τ =.99 τ kriticke = τ =. τ kriticke =. τ =. τ kriticke =

23 τ =.9 τ kriticke =. τ =.99 τ kriticke = τ =. τ kriticke =. τ =. τ kriticke =

24 τ =.9 τ kriticke =. τ =.99 τ kriticke = τ =. τ kriticke =. τ =. τ kriticke =

25 The explicit method is weakly stable if τ h a ; strongly stable if τ < h a.

26 Implicit stencil U k+ i Ui k + U k i τ [ = U k+ a i Uk+ i + U k+ i+ h + Uk i Uk i h + U k i+ Seven-point stencil If the approximate solution is known at the kth time-level, then we have to establish and solve a tridiagonal system of linear algebraic equations to get to the next time level. The method is stable for any τ >, i.e., unconditionally stable. ].