FDM for wave equations

Transcription

1 FDM for wave equations Consider the second order wave equation Some properties Existence & Uniqueness Wave speed finite!!! Dependence region Analytical solution in 1D

2 Finite difference discretization

3 Finite difference discretization Discretization Explicit method

4 Finite difference discretization Local truncation error Consistency: yes!! Stability condition see details in class or as an exercise Convergence rate based on Lax equivalence theorem

5 FDM for wave equation u 2u + u Implicit scheme exercise n+ 1 n n 1 2 j j j ν θ n+ n+ n+ n n n n n n uj 1 uj uj 1 θ uj 1 uj uj 1 θ = uj+ 1 uj + uj 1 k h [ ( 2 ) (1 2 )( 2 ) ( 2 )] Efficient computation & error estimates Extension to 2D & 3D cases With variable coefficients & with other BCs

6 For nonlinear wave equation Nonlinear wave equation d u( xt, ) uxt (, ) + f( u) = 0, x, t> 0 tt ux (,0) = φ( x), ux (,0) = ϕ( x), x Sine-Gordon equation: Klein-Gordon equation: Soliton solution & energy conservation t f( u) = sin( u) f( u) = u+ gu ( ) Et = uxt + uxt + Fuxt dx E 2 2 () t (,) (,) ((,)) (0) d u 2 2 : = ϕ( ) + φ( ) + ( φ( )), ( ) = ( ) d x x F x dx F u f s ds 0 d

7 For nonlinear wave equation Numerical methods Semi-implicit scheme n+ 1 n n 1 uj 2uj + uj 1 n+ 1 n+ 1 n+ 1 n 1 n 1 n 1 n u 2 2 j+ 1 2uj + uj 1 + uj+ 1 2 uj + uj 1 + f( uj) = 0 k 2h Energy conservative scheme u 2 u + u Fu ( ) Fu ( ) = 0 n+ 1 n n 1 n+ 1 n 1 j j j 1 n+ 1 n+ 1 n+ 1 n 1 n 1 n 1 j j u 2 2 j 1 uj uj 1 uj 1 uj uj 1 n 1 n 1 k 2h uj uj Nonrelativistic limit regime -- current research topics!!! 2 1 ε tt ( xt, ) uxt (, ) + u+ f( u) = 0, 0< ε 1 2 ε

8 For nonlinear Schrodinger equation The nonlinear Schr o dinger equation (NLS) 1 i t (,) xt = + V( x) ψ ψ ψ β ψ ψ t : time & : spatial coordinate ψ (,) xt V( x ) : complex-valued wave function : real-valued external potential β : coupling constant =0: linear d x ( R ) >0: repulsive interaction <0: attractive interaction

9 Modelling In quantum physics & nonlinear optics: Interaction between particles with quantum effect Bose-Einstein condensation (BEC): bosons at low temperature Superfluidity Propagation of laser beams, In plasma physics: wave interaction between electrons and ions Zakharov system,.. In quantum chemistry: chemical interaction based on the first principle Schr o dinger-poisson system

10 Modelling In particle physics: interaction between fundamental particles Klein-Gordon-Schr o dinger (KGS) system Maxwell-Dirac (MD) system In biology: protein folding, Schr o dinger-poisson system,.. In materials science: First principle computation Semiconductor industry..

11 Numerical difficulties for dynamics Dispersive & nonlinear Solution and/or potential are smooth but may oscillate wildly Keep the properties of NLS on the discretized level Time reversible & time transverse invariant Mass & energy conservation Dispersion relation In high dimensions: many-body problems Design efficient & accurate numerical algorithms Explicit vs implicit (or computation cost) Spatial/temporal accuracy, Stability Resolution in strong interaction regime: β 1

12 Algorithms for computing dynamics Different methods Crank-Nicolson finite difference method (CNFD) Time-splitting spectral method (TSSP) Leap-frog (or RK4) + FD (or spectral) methods.. Time-splitting spectral method (TSSP) i tψ( xt, ) = ( A+ B) ψ with A=, B= V( x) + β ψ 2 ia t ib t 2 e e ψ ( xt, n) + O(( t) ) i( A+ B) t ia t/2 ib t ia t/2 3 ψ( xt, n+ 1) = e ψ( xt, n) e e e ψ( xt, n) + O(( t) ) 5 + O(( t) )

13 Algorithms for computing dynamics For, apply time-splitting technique [ tn, t n + 1] Step 1: Discretize by spectral method & integrate in phase space exactly 1 2 i tψ(,) xt = ψ 2 Step 2: solve the nonlinear ODE analytically 2 i tψ(,) xt = V( x) ψ(,) xt + β ψ(,) xt ψ(,) xt = = i ψ xt = V xψ xt + β ψ xt ψ xt t 2 t( ψ( xt, ) ) 0 ψ( xt, ) ψ( xt, n) 2 (, ) ( ) (, ) (, n) (, ) 2 i( t tn)[ V( x) + βψ ( xt, n) ] ψ( xt, ) = e ψ( xt, n) Use 2 nd order Strang splitting (or 4 th order time-splitting)

14 An algorithm for the dynamics of NLS ψ ψ ψ ( x) = e ψ ( x) ( x) = F e F( ψ ) ( x) = e ψ ( x) n 2 (1) i tv [ ( x) + βψ ( x) ]/2 n (2) 1 ia t (1) (2) 2 n+ 1 i tv [ ( x) + βψ ( x) ]/2 (2) Different spectral transforms for F: Fourier transform, Laguerre transform, Hermite transform, Gaussian transform, Different ways of transform (1) (2) ( d) For small d: ψ( x1, x2,..., xd) = ψ j1, j2,..., jφ ( 1 1) ( 2 2)... ( ) d j x φj x φj x d d N 1 j1, j2,..., jd N For large d: d ψ( x, x,..., x ) = ψ φ ( x ) φ ( x )... φ ( x ) N(log( N)) (1) (2) ( d) 1 2 d j1, j2,..., jd j1 1 j2 2 jd d j j... j N 1 2 d d

15 Properties of the method Explicit & computational cost per time step: Time reversible: yes Time transverse invariant: yes Mass conservation: yes Stability: yes Dispersion relation without potential: yes Accuracy Spatial: spectral order; Temporal: 2 nd or 4 th order OM ( ln M) Best resolution in strong interaction regime: β 1

16 For hyperbolic conservation laws Consider 1D conservation law Where f is a convex function Three types of problems Cauchy problem -- defined in the whole space Riemann problem initial data has discontinuity Initial-boundary problem

17 Special cases Linear case, i.e. f(u) = a u Exact solution Burgers equation It can generate shock solution for smooth initial data

18 Special cases Riemann problem Shock solution Shock speed satisfies the Rankine-Hugoniot condition Rarefaction solution

19 Properties Conservation Characteristics Line for linear case Curve for nonlinear problem Dependence region Wind direction Shock formation

20 Finite difference discretization Numerical difficulties Discontinuous solution Keep physical laws in the discretized level Boundary condition setups must according to characteristics.. Numerical methods Shock tracking random selection method Shock capturing Upwind method, Lax-Friedrichs method,. Total variation diminishing (TVD) schemes, Godunov scheme, ENO or WENO scheme Gas-kinetic scheme, relaxation scheme,

21 Finite difference discretization

22 A fake method Forward Euler + 2 nd central finite difference Stability von Neumann method Assume Plugging into the difference equation Amplification factor It is unconditionally unstable!!!

23 Numerical method Lax-Friedrichs method For linear problem, e.g. f(u) = 3u For Burgers equation Convergence Local truncation error -- consistency Conservation: yes!!

24 Numerical method CFL (Courant-Friedrichs-Lax, 1928) condition--- stability For linear problem For nonlinear case TVD property Total variation Lax-Friedrichs is a TVD scheme under the stability condition Convergence Lax equivalent theorem

25 High resolution scheme Godunov (Godunov, 1959) scheme Use the conservation of the equation & (approximate) Riemann solver

26 High resolution scheme Integrate exactly Conservative scheme

27 Second-order method

28 Second-order method Choice of slope limiter Roe s superbee limiter Van Leer limiter..

29 Examples

30 Numerical methods Properties of Godunov schemes Local truncation error: Consistency: yes!! Explicit, conservative & TVD method CFL (stability) condition: Convergent under the stability condition More topics High dimensions dimension-by-dimension System of conservation laws Euler system, shallow water,...

31 Finite element method FEM for Poisson equation Variational formulation Sobolev spaces: { x } j 1 1 { φ( ) φ ( ), φ Γ 0 }, g { φ( ) φ ( ), φ Γ } Variational problem ux ( ) = f( x) in Ω ux ( ) = gx ( ) on Γ = Ω (P1) L ( Ω ) = φ( x) φ( x) dx <, ( φϕ, ) = φ( x) ϕ( x) dx, φ = ( φφ, ) 2 2 1/2 2 L Ω Ω H x L L j d ( Ω ) = φ( ) φ ( Ω), φ ( Ω ), = 1,...,, φ = φ + φ, φ : = φ V = x H Ω = V = x H Ω = g H L L H L Find u V such that auv (, ) = f( v) v V g a( u, v) = u( x) v( x) dx, f ( v) = f ( x) v( x) dx (P2) Ω Ω 0

32 Finite element method Minimization formulation Theorem: Find u V such that (P3) 1 Eu ( ) = min Ev ( ) Ev ( ) = avv (, ) f( v) vv g 2 Partition of the domain In 1D: intervals g In 2D: triangles or rectangles or quadrilaterals, etc. In 3D: tetrahedrons, etc. If u C ( P1) ( P2) ( P3) ( P1) T : a = x < x < < x = b, T = [ x, x ], h = x x, h = max h h 0 1 N i i i+ 1 i i+ 1 i i 0 i N 1 2

33 Finite element method Finite element approximation Finite element subspaces { φ( ) ( ) φ ( ), 0,1,..., 1; φ( ) αφ, ( ) β} i { φ ( ) ( ) φ T ( i), 0,1,..., 1; φ ( ) φ ( ) 0} V = x C Ω P T i = M a = b = h h h k h h g T i V = x C Ω P T i = M a = b = h h h k h h 0 Finite element approximation Find u h Error Estimates (Cea lemma) h 0 i V h g such that au v = f v v V h h h h h h (, ) ( ) 0 (P ) u u C u v C u u Ch v = a v v V V h h h k 1/2 h h inf π, (, ), π : V h V V V g g v V

34 Finite element method How to form the linear system (for k=1 as example) Total stiff matrix & load vector h h h h h h h h T a( u, v ) = u ( x) v ( x) dx = u ( x) v ( x) dx : = u ( x) v ( x) dx = V AU Ω h h h h T f( v) = f( xv ) ( xdx ) = f( xv ) ( xdx ) : = f( xv ) ( xdx ) = V F Ω i h i Element stiff matrix & load vector i h T T T T T T T T T T i u a u v u x v x dx v v N x N x dx V AU xi+ 1 xi+ 1 h h h h T i T T(, ): = ( ) ( ) ( 1) ( ) ( ) : i x x = i i+ x i x i i i i u = x x i+ 1 i xi+ 1 xi+ 1 h h T xi+ 1 x x xi T( ): ( ) ( ) ( 1) ( ) ( ) ( ) i i i+ i i i i x x x i+ 1 xi xi+ 1 xi f v = f xv xdx= v v f xn xdx= V F N x = i i i i h x i+ 1 x i i h x i+ 1 x i T

35 Finite element method First compute element stiff matrix & load vector Assembling the total stiff matrix & load vector Plugging into the boundary condition How to solve the linear system: spare & large PCG or GMRES Multigrid,.. FEM software packages: ANSTRAN: ADIAN: More: