Numerical Oscillations and how to avoid them

Transcription

1 Numerical Oscillations and how to avoid them Willem Hundsdorfer Talk for CWI Scientific Meeting, based on work with Anna Mozartova (CWI, RBS) & Marc Spijker (Leiden Univ.) For details: see thesis of A. Mozartova [Nov. 26, 22]. /2

2 Numerical methods in MAC3 Numerical methods in the MAC3 group are and used for simulations, to understand physical phenomena; used for simulations, to enhance analysis; studied as mathematical objects (analysis of numerical methods): to understand the behaviour of methods, to design more efficient methods. This talk is about the third item, the analysis of numerical methods. In particular time stepping methods for partial differential equations (PDEs) with dominating convection. 2/2

3 Example of simulation: growth of plasma channels Expensive simulations, need for efficient methods: σ 4 2 σ σ Fig: Streamer formation and branching. Electron densities at time t = 2 (left), t = 4 (middle) and t = 6 (right). 3/2

4 Convection-dominated problems Typical problem: convection-diffusion-reaction, u = u(x, t), t u + f (u) = (κ(u) u) + ǫ g(u), in 2D or 3D (x Ω R 2 or R 3 ). In this talk, for simplicity: D conservation laws t u + f (u) =. x Spatial discretization on grid {x j } with x j = j x, u j (t) u(x j, t). Simple example: st order upwind (for f (u) > ) u j(t) = ( ) f (u j ) f (u j ) x (for j =, 2,...,m). Then: system of ordinary differential equations (ODEs) in R m : u (t) = F(u(t)). Efficient and reliable time stepping methods are needed! 4/2

5 Example: shallow water equations Shallow water equations: water height h = h(x, t) and water velocity v = v(x, t), ( ) h + ( ) hv t hv x hv 2 + =. 2 gh2 Here g is gravity constant, 5 x 5 with periodic boundary conditions and initial conditions: Water Height 2 { h(x, ) = 2 if 4 x ;.5 h(x, ) = otherwise; v(x, ) =.2. Water Velocity Matlab simulation: time evolution of h and v, with space variable x on horizontal axis, x = /, various time steps t. 5/2

6 Numerical solutions at t = 2.5 with t x =.: Water Height Water Velocity Very slow! 6/2

7 Numerical solutions at t < 2.5 with t x =.5: Water Height 5 5 Water Velocity Negative h, end-time not reached, overflow! 7/2

8 Numerical solutions at t = 2.5 with t x =.45: Water Height Water Velocity No negative values for h, no overflow. 8/2

9 Central questions: How large can we take the time step t in such a problem? What are good time stepping methods for such problems? For this last item, we want a good qualitative behaviour near discontinuities, but also high accuracy in regions where the solution is smooth. A time stepping method with error t p for smooth problems is said to have order p. 9/2

10 Linear Multistep Methods ODE system u (t) = F(u(t)), approx. u n u(t n ), t n = n t. Linear multistep (LM) method, with coefficients a j, b j, u n = k k a j u n j + b j tf(u n j ) j= j= (n k). Either explicit (b = ) or implicit (b > ). Examples: implicit BDF2 and extrapolated BDF2, k = p = 2 : (Ex) u n = 4 3 u n 3 u n tf(u n ) 2 3 tf(u n 2) for n 2. First step: explicit Euler u = u + tf(u ). (Im) u n = 4 3 u n 3 u n tf(u n) for n 2. First step: implicit Euler u = u + tf(u ). This implicit method is unconditionally stable, allowing large stepsizes t for smooth problems. /2

11 Theoretical framework: monotonicity and boundedness Consider the ODE system u (t) = F(u(t)) on R m with given initial value. Basic assumption: (A) v + τ F(v) v (for all v). This basic assumption holds for scalar, D conservation laws with suitable spatial discretization, τ x, and maximum-norm v = max j v j, total variation semi-norm v TV = j v j v j. Similarly with more general sublinear functionals, to include e.g. positivity preservation. /2

12 We study following property of the multistep methods, (B µ ) sup u n µ max u j whenever (A) and t γτ n k j<k The step-size coefficient γ > and boundedness factor µ only depend on the method (not on the problem). µ = : monotonicity (also known as TVD for TV-seminorm), µ > : boundedness (also known as TVB for TV-seminorm). The monotonicity property, with µ =, is too strict; it excludes many good methods, such as BDF and Adams methods. The TVB (total variation boundedness) property guarantees convergence towards the physically correct solution. 2/2

13 Boundedness for LMMs More relaxed conditions with boundedness factor µ >. Define ρ j = for j <, ρ = and ρ n = k j= a jρ n j γ k j= b jρ n j (n ), π n = k j= b jρ n j (n ). Theorem Let γ >. Then (under minor technical assumptions) the boundedness property (B µ ) is valid for some µ if and only if π n for all n. The sequences ρ n, π n are determined by the behaviour of the method for the very simple test equation u = λu, tλ = γ. 3/2

14 Monotonicity with starting procedures Suppose the starting values u,...,u k are computed from the given u by a starting procedure (e.g. a Runge-Kutta method of order p ). More reasonable than arbitrary u,...,u k. Then with suitable starting procedure we can still have the monotonicity property sup u n u whenever (A) and t γτ. n k The necessary and sufficient criterion for this is the boundedness condition π n for all n, together with inequalities k c j ρ n j (n ) j= for some tuples c = (c, c,...,c k ), c j = c j (γ) determined by the starting procedure. 4/2

15 Example: Implicit and extrapolated BDF2 for Buckley-Leverett equation Buckley-Leverett equation: u t + f (u) x =, f (u) = u 2 u 2 +, 3 ( u)2 with u(, t) = 2 and initial block-function (zero on (, 2 ], one on ( 2, ]). Flux-limited spatial discretization (van Leer type); fixed grid with x = 5 3. t =.8.6 t = / /2

16 Extrapolated BDF2 scheme: u n = 4 3 u n 3 u n tf(u n ) 2 3 tf(u n 2) (n 2). Order 2; stable for Courant numbers t x max f (u) 2. First step: explicit Euler u = u + tf(u ). Stepsize coefficient: γ = 5 8. Implicit BDF2 scheme: u n = 4 3 u n 3 u n tf(u n) (n 2). Order 2; unconditionally stable. First step: implicit Euler u = u + tf(u ). Stepsize coefficient: γ = 2. 6/2

17 Plots of numerical solutions at time t = 4 with t/ x = / extrap. BDF impl. BDF /2

18 Plots of numerical solutions at time t = 4 with t/ x = /8, t/ x = / extrap. BDF impl. BDF /2

19 Plots of numerical solutions at time t = 4 with t/ x = /8, t/ x = /4, t/ x = / extrap. BDF impl. BDF /2

20 Conclusions/Outlook For multistep methods: the standard monotonicity theory is too strict; better to consider boundedness for arbitrary starting values, or monotonicity with suitable starting procedure. For problems with diffusion or stiff reactions, some implicitness of the method is needed. This can be done with the so-called combined implicit-explicit (IMEX) methods, taking convection explicit and diffusion/reactions implicit. The IMEX multistep perform well in numerical tests. However, proper theory is lacking. An automatic IMEX multistep code has not yet been developed! 2/2