1 Upwind scheme for advection equation with variable. 2 Modified equations: numerical dissipation and dispersion

Transcription

1 1 Upwind sceme for advection equation wit variable coefficient Consider te equation u t + a(x)u x Applying te upwind sceme, we ave u n 1 = a (un u n 1), a 0 u n 1 = a (un +1 u n ) a < 0. CFL condition is a / 1. Te von Neumann analysis is not appropriate since te coefficients are not constant. Te energy metod for l 2 stability is so obvious. (Tere is some energy estimate for te leapfrog metod.) Here, we can do l analysis. Te metod can be written as = (1 a )un + a un, = 1, a > 0 and = + 1, a < 0. Hence, (1 a ) un + a un = u n. 2 Modified equations: numerical dissipation and dispersion Code presentation: In tis example, we solve u t +u x = 0 numerically wit initial data u 0 (x) = exp( 20(x 2) 2 ) + exp( (x 5) 2 ). We ll compare te upwind sceme and Lax-Wendroff sceme. To solve tis problem, we truncate te domain to [0, 20] up to time t = 10. Note tat for te time interval we considered, u(0, t) 0 (for better approximation, you can use u(0, t) = exp( 20(t+2) 2 )+exp( (t+5) 2 )). For te rigt boundary, we use one-sided approximation for te finite difference. It s clear tat te Lax-Wendroff is more accurate. If we loo at te beavior of te scemes more closely, we find tat te upwind sceme tends to smoot out te corners and it as some dissipating (diffusion) effect. Te Lax-Wendroff owever causes oscillation near te corners, wic suggests tat te Lax-Wendroff as te dispersion effect. 1

2 2.1 Analysis using modified equations Compared wit te original PDE, it s possible to find a PDE tat is better satisfied by our numerical metod. Tese PDEs are called te modified equations. Consider te upwind sceme for u t + au x = 0 a > 0: u n = a (un u n 1). Suppose v(x, t) is a smoot function tat satisfies tis numerical metod exactly. Ten, we ave v(x, t n+1 ) v(x, t n ) = a (v(x, t n ) v(x 1, t n )). v t (x, t n )+ 1 2 v tt(x, t n )+O( 2 ) = a[v x (x, t n ) 1 2 v xx]+o( 2 ) Hence, Tis suggests v t + av x = a 2 v xx 1 2 v tt + O( ). v tt = av xt + O( + ) = a 2 v xx + O( + ). Inserting tis into te term above, we ave v t + av x = a 2 ( a)v xx + O( ) Tis means tat v(x, t) satisfies te equation v t + av x = a 2 ( a)v xx better tan v t + av x Te modified equation is advection-diffusion equation. Clearly, if a > 0 or a < 1, tere is diffusion effect. Tis is called te numerical diffusion. Similarly, te modified equation for Lax-Wendroff is v t + av x = 1 6 a2 (1 ( a )2 )v xxx + O( ) Te main error for te transport equation is v xxx. If you compute te dispersion relation: iω + aiξ = 1 6 a2 (1 ( a )2 )( i)ξ 3 ω = aξ 1 6 a2 (1 ( a )2 )ξ 3. Hence, te main error term is dispersive and tat is wy te oscillation appears tere. 2

3 Comment: Te numerical dispersion relation for te FDM (or equivalently, te dispersion relation for te full modified equation) can be computed by inserting u n = exp(i(ξx ωt n )) into te finite difference metod. If you compare tis wit von-neumann, e iω(ξ) = g(ξ). 2.2 Adding numerical dissipation/diffusion Sometimes, we desire to ave numerical diffusion to smoot out numerical solutions and damp some useless modes. Consider te Leapfrog metod: u n 1 2 = a un +1 un 1. 2 Let s compute te exact numerical dispersion relation. Assuming v(x, t) = exp(i(ξx ωt)) satisfies tis numerical metod exactly. Ten, we ave sin(ω) = a sin(ξ). Ten, we see tat ω is a real function of ξ. Hence, for every order of and, te Leapfrog metod only as dispersion and tere is no dissipation. Tat means all modes in te error for te advection equation will not damp. Hence, if one wants to damp tose modes, we can add numerical dissipation. One possible way is to solve te following FDM: u n 1 2 = a 1 2 (un +1 u n 1) ɛ 4 2 (1 4 D2 ) 2 u n 1 If you compute te numerical dispersion relation, you ll see tat te added term gives dissipation. 3 Second order yperbolic equations: wave equation Consider te wave equation u(x, 0) = f(x), u tt = a 2 u xx, u t (x, 0) = g(x). 3

4 Note if we introduce v = u t, w = au x, we ave te first order system of equations Here, we use te simplest sceme: 2u n + un 1 2 v t aw x = 0 w t av x a 2 un +1 2un + un 1 2 Te boundary condition u t can be approximated by te gost point metod. First of all, let s consider te CFL condition. Te PDE as two caracteristic speeds. One is dx/dt = a and one is dx/dt = a. Ten, te CFL condition implies tat a. Let s derive te accurate condition using von-neumann analysis, wic is a < 1. One can assume u n = g(ξ)n e ix ξ to get te amplification factor. Here, let s sow te matrix formulation. Introducing v n = un un 1, w+1/2 n = aun +1 un. Te te sceme can be rewritten as v n+1 v n a wn +1/2 wn 1/2 v 1 n+1 w n+1 1/2 wn 1/2 a vn+1 = 0 By doing so, we ave a two-time level sceme. Ten, v n = v n e ixξ and w n = wn e ixξ. Ten, we ave Letting c = 2a v n+1 v n w n+1 w n sin( 1 2ξ), we ave a wn 2i sin( 1 2 ξ) a vn+1 2i sin( 1 2 ξ) v n+1 = v n + icw n w n+1 = icv n + (1 c 2 )w n. = 0 4

5 Te e-vals satisfy µ 2 (2 c 2 )µ + 1 We need µ 1. Te product is 1 ence bot eigenvalues must ave magnitude 1. (2 c 2 ) = µ+µ 1 = 2 cos(θ). Hence, we need c 2 4 a 1. However, te norm of te matrix may grow if te eigenvalues are repeated. Tis appens if c = 0 or c 2 = 4. c = 0 is fine since te matrix as two Jordan blocs. c = ±2 corresponds to a/ = 1. Tis is bad since te matrix as only one Jordan bloc. Te metod is unstable. Hence, te stability condition is a/ < 1 (unlie te advection equation). 4 Te yperbolic systems For 1D yperbolic system u t + Au x (A is diagonalizable and as real e-values), te Lax-Friedrics, Lax-Wendroff and Leapfrog scemes can be generalized easily. Te stability condition sould be satisfied for all eigenvalues. Te non-trivial generalization is for upwind sceme and Beam-Warming. If eigenvalues are all nonnegative or all are nonpositive, te generalization is also easy. Te issue arises if some are positive wile some are negative. In tis case, we sould find te caracteristic variables (i.e. te eigenvectors). For tose wit positive e-values, we use te a > 0 upwind and for tose wit negative e-vals, we use a < 0 upwind. 5 Nonlinear yperbolic conservation laws & multidimension problems Te Finite Volume Metods are more suitable for nonlinear yperbolic conservation laws. We ll come bac later. For multi-dimension problems, you can use LOD or ADI ideas to design scemes. We ll not tal about tem in our class. Tose wo are interested can read more references. 5