VIEWS: 218 PAGES: 36 POSTED ON: 8/13/2010 Public Domain
FDM for parabolic equations Consider the heat equation – where Well-posed problem – Existence & Uniqueness – Mass & Energy decreasing FDM for parabolic equations CNFD Crank-Nicolson + 2nd order finite difference Questions – How to solve the equations efficiently??? – Convergence and order of accuracy??? • Local truncation error & Stability Local truncation error Linear system Order of accuracy: 2nd in space and time Consistency: yes!!! Linear system – With Implicit scheme!!! – At each time step, we need solve a linear system Matrix form Solution algorithm Convergence analysis Convergence – Consistency & Stability Consider the general problem u( x , t ) L( u ) (0, T ) L differential operator t g (u ) g 0 u( x ,0) u0 ( x ) It is a well-posed problem: – Existence, uniqueness, continuously depend on initial data Finite difference discretization Time step: k t tn n k , n 0,1, 2, , N Mesh size: h x Index set of grid points: J Exact solution at level n: u n ( x ) : u( x , tn ) u n ( x ) 2 : |u n ( x ) |2 d x Exact solution vector at level n on grid points: un : {u( x j , tn ), j J } FDM approximation vector at level n: U n : {un , j J } j Norms – Maximum norm U n : max {| u n |, j J } j – 2-norm U n : | u n |2 V j j with V j weights 2 jJ Finite difference discretization General form of finite difference scheme B1 U n 1 B0 U n F n with B1 & B0 difference operators – Assume B1 is invertible, i.e. its representing matrix is non-singular n 1 1 U B [ B0 U F ] 1 n n – Formally it represents the differential equation in the limit n 1 h ,k 0 u B1 U [ B0 U F ] n n L(u) 0 B1 O(1 / k ) t – Uniformly well-conditioned B11 K1 t for all h h0 & k k0 Convergence analysis Truncation error: T n : B1 u n 1 [ B0 u n F n ] with u n exact solution Consistency: Tjn 0 when k t & h x 0 for all j J Order of accuracy: p-th order in time & q-th order in space | Tjn | C[k p hq ] when k t & h x 0 for all j J Convergence U n u n h , 0 0 with n k t (0, T ] k Order of convergence: p-th order in time & q-th order in space U n un C[k p h q ] when k & h 0 Stability: V n W n K V 0 W 0 for all nk T – two solutions have the same inhomogeneous terms but start with difference initial data Convergence analysis Stability condition: ( B11B0 )n 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!! Von Neumann method for stability For CNFD Plugging into CNFD Amplification factor Unconditionally stable—no constraint for time step!!!!! Energy method – See details in class or as an exercise!! 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!! Method of line approach Discretize in space first Method of line approach Method of line approach An ODE system Discretize in time by ODE solver – Trapezoidal method – Forward Euler method – Backward Euler method – Runge-Kutta method, …….. Method of line approach Discretize in time first Method of linear approach Discretize in space by finite difference – This is CNFD – Other discretization in space is possible 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 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 Extension For Neumann BC Discretization: CNFD Extension Local truncation error: 2nd order in space & time Consistency: yes!! Implicit method Linear system -- exercise Matrix form – exercise Stability: unconditionally stable!! Convergence: Extension Variable coefficients Discretization -- CNFD Extension Local truncation error: 2nd order in space & time Consistency: yes!! Implicit method Linear system -- exercise Matrix form – exercise Stability: unconditionally stable!! Convergence: Extension 2D heat equation Discretization – Crank-Nicolson in time – Second order central difference in space Discretization Extension Local truncation error: 2nd order in space & time Consistency: yes!! Implicit method Linear system – At every step, use direct Poisson solver Matrix form – exercise Stability: unconditionally stable!! Convergence: 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 Nonlinear parabolic PDEs Allen-Cahn equation t u u u (1 u 2 ) (0, T ) u( x , t ) 1 u( x ,0) u0 ( x ) Applications – Imaging science – Materials science – Geometry,…… 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 Time-splitting method From [tn , tn 1 ] , apply time-splitting technique – Step 1. Solve nonlinear ODE for half-step—integrate exact!!! t u u (1 u 2 ) – Step 2. Solve a linear PDE for one step-- CNFD t u u – Step 3. Solve nonlinear ODE for half-step — Integrate exact!!!! t u u (1 u 2 ) Accuracy in time: second order!!!! No need to solve nonlinear system!!!! Implicit-explicit method Ideas: – Implicit for linear terms & Explicit for nonlinear terms Discretization – Method 1 – for computing dynamics u n 1 u n 1 3 n 1 n 1 [ h u n 1 h u n ] u n 1/2 (1 (u n 1/2 )2 ), u n 1/2 : u u k 2 2 2 – Method 2 – Convex-concave splitting u n 1 u n 1 1 [ h u n 1 h u n 1 ] (u n 1 u n 1 ) u n u n (1 (u n ) 2 , 0 2k 2 2 – Method 3 – for computing steady state u n 1 u n h u n 1 u n 1 u n [ (1 (u n ) 2 )] parameter k Integrate factor (IF) method Rewrite t u u u (1 u ) 2 Multiply both side t ( e t u ) [ t u u ]e t e t u (1 u 2 ) Integrating over [tn , tn 1 ] tn 1 u ( tn ) tn 1 tn e u(tn 1 ) e e t u(t ) (1 u 2 (t ))dt tn tn 1 u(tn 1 ) e u(tn ) e k ( tn 1 t ) u(t ) (1 u 2 (t ))dt tn Approximate in time via RK4 & in space via FDM un1 ekh un S (ekh un (1 (un )2 )) j j j j Nonlinear parabolic PDEs Sharp interface 1 t u u u (1 u 2 ) 0 1 2 Ginzburg-Landau equation (GLE) 1 t u u u (1 | u |2 ) u: d 2 General nonlinearity t u u f (u ) System,....... Compact scheme in space 1 | u |2 ) u: d 2 General nonlinearity t u u f (u ) System,....... Compact scheme in space