FDM for parabolic equations

Transcription

1 FDM for parabolic equations Consider the heat equation where Well-posed problem Existence & Uniqueness Mass & Energy decreasing

2 FDM for parabolic equations

3 CNFD Crank-Nicolson + 2 nd order finite difference Questions How to solve the equations efficiently??? Convergence and order of accuracy??? Local truncation error & Stability

4 Local truncation error

5 Linear system Order of accuracy: 2 nd in space and time Consistency: yes!!! Linear system With Implicit scheme!!! At each time step, we need solve a linear system

6 Matrix form

7 Solution algorithm

8 Convergence analysis Convergence Consistency & Stability Consider the general problem uxt (,) = Lu ( ) Ω (0, T) L differential operator t gu ( ) = g0 Γ = Ω ux (,0) = u( x) Ω 0 It is a well-posed problem: Existence, uniqueness, continuously depend on initial data

9 Finite difference discretization Time step: k= t tn = nk, n= 0,1, 2,, N Mesh size: h = x Index set of grid points: Exact solution at level n: n Exact solution vector at level n on grid points: n u : = { ux ( j, tn), j J Ω } n FDM approximation vector at level n: Norms J Ω n n n 2 u ( x): u( x, t ) u ( x) : u ( x) d x = = 2 Ω n U : = { uj, j J Ω } n n Maximum norm U : = max { u, j J } 2-norm j Ω n n 2 U : = u V with V weights 2 j J Ω j j j

10 Finite difference discretization General form of finite difference scheme BU + = BU + F with B& B difference operators n 1 n n Assume B1 is invertible, i.e. its representing matrix is non-singular U = B [ BU + F] n+ 1 1 n n 1 0 Formally it represents the differential equation in the limit hk, 0 n+ 1 n n u BU [ BU + F ] Lu ( ) = 0 B = O(1 / k) t Uniformly well-conditioned B K t for all h h & k k

11 Convergence analysis Truncation error: Consistency: Order of accuracy: p-th order in time & q-th order in space Convergence Order of convergence: p-th order in time & q-th order in space Stability: T : = Bu [ Bu + F] with u exact solution n n+ 1 n n n 1 0 n Tj 0 when k = t& h = x 0 for all j J Ω n p q Tj Ck [ + h] when k= t& h= x 0 for all j J Ω hk, 0 n n U u 0 with nk t (0, T] U n u n Ck [ p + h q ] when k& h 0 V W KV W n n 0 0 for all nk two solutions have the same inhomogeneous terms but start with difference initial data T

12 Convergence analysis Stability condition: n ( B B ) K for all nk T von Neumann method based on Fourier transform Energy method Lax Equivalence Theorem: For a consistent difference approximation to a well-posed linear evolutionary problem, which is uniformly well-conditioned, the stability of the scheme is necessary and sufficient for the convergence. Proof: See details in class or as an exercise!!

13 Von Neumann method for stability

14 For CNFD Plugging into CNFD Amplification factor Unconditionally stable no constraint for time step!!!!! Energy method See details in class or as an exercise!!

15 Convergence analysis Convergence of CNFD Consistency Unconditionally stable From Lax equivalent theorem implies convergence!!! Convergence rate Other methods for analysis Energy method -- Exercise!! Based on maximum principle Exercise!!

16 Method of line approach Discretize in space first

17 Method of line approach

18 Method of line approach An ODE system Discretize in time by ODE solver Trapezoidal method Forward Euler method Backward Euler method Runge-Kutta method,..

19 Method of line approach Discretize in time first

20 Method of linear approach Discretize in space by finite difference This is CNFD Other discretization in space is possible

21 Other discrtization for heat equation Forward Euler finite difference method Local truncation error: Explicit method & direct marching in time Consistency: yes!! Stability condition: Under stability condition, it converges

22 Other discrtization for heat equation Backward Euler finite difference method Local truncation error: Implicit method: At each step, the linear system can be solved by Thomas algorithm Consistency: yes!! Unconditionally stable!!! It converges and has convergence rate

23 Extension For Neumann BC Discretization: CNFD

24 Extension Local truncation error: 2 nd order in space & time Consistency: yes!! Implicit method Linear system -- exercise Matrix form exercise Stability: unconditionally stable!! Convergence:

25 Extension Variable coefficients Discretization -- CNFD

26 Extension Local truncation error: 2 nd order in space & time Consistency: yes!! Implicit method Linear system -- exercise Matrix form exercise Stability: unconditionally stable!! Convergence:

27 Extension 2D heat equation Discretization Crank-Nicolson in time Second order central difference in space

28 Discretization

29 Extension Local truncation error: 2 nd order in space & time Consistency: yes!! Implicit method Linear system At every step, use direct Poisson solver Matrix form exercise Stability: unconditionally stable!! Convergence:

30 More topics With Rabin or periodic BCs 2D heat equation in a disk or a shell 3D heat equation in a box, spehere,. More general case ADI (alternating direction implicit) for 2D & 3D Compact scheme Nonlinear equation & system of heat equations

31 Nonlinear parabolic PDEs Allen-Cahn equation 2 t u = u+ λ u(1 u ) Ω (0, T) uxt (,) = 1 Γ = Ω ux (,0) = u0( x) Ω Applications Imaging science Materials science Geometry,

32 Numerical methods Standard finite difference methods Crank-Nicolson finite difference Forward Euler finite difference Backward Euler finite difference Special techniques Time-splitting (split-step) method Implicit-explicit method Integration factor method

33 Time-splitting method From, apply time-splitting technique [ tn, t n + 1] Step 1. Solve nonlinear ODE for half-step integrate exact!!! t u = λ u u 2 (1 ) Step 2. Solve a linear PDE for one step-- CNFD t u = Step 3. Solve nonlinear ODE for half-step Integrate exact!!!! Accuracy in time: second order!!!! No need to solve nonlinear system!!!! u 2 (1 ) t u = λ u u

34 Implicit-explicit method u n+ 1 u k u Ideas: Implicit for linear terms & Explicit for nonlinear terms Discretization n n+ 1 n 1 u Method 1 for computing dynamics = [ h u + u ] + λ u (1 ( u ) ), u : = u u n+ 1 n n+ 1/2 n+ 1/2 2 n+ 1/2 n n 1 h Method 2 Convex-concave splitting u = k 2 2 Method 3 for computing steady state n+ 1 u k n 1 [ n+ 1 n 1 α ] ( n+ 1 n 1 ) n n (1 ( n ) 2 h u h u u u αu λ u u, α 0 n+ 1 n+ 1 n n 2 = h u αu + u α + λ u α [ (1 ( ) )] parameter

35 Integrate factor (IF) method Rewrite Multiply both side t u u = λ u u 2 (1 ) t t t 2 t( e u) = [ t u ue ] = λe u(1 u) Integrating over [ tn, t n + 1] n+ 1 tn+ 1 e ut ( tn ) e ut ( ) = λ t 2 e ut ()(1 u()) t dt n+ 1 n+ 1 ( ) = k ( ) + λ ( tn+ 1 t) ()(1 2 ()) n+ 1 n t n ut e ut e ut u t dt t n t t Approximate in time via RK4 & in space via FDM n u + = e u + λse ( u (1 ( u) )) n 1 k h n k h n n 2 j j j j

36 Nonlinear parabolic PDEs Sharp interface 1 t u = u+ u u < ε 2 ε Ginzburg-Landau equation (GLE) 2 ε General nonlinearity u = u+ f( u) t System,... Compact scheme in space 2 (1 ) d t u = u+ u(1 u ) u: