Time stepping methods
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1 Time stepping methods ATHENS course: Introduction into Finite Elements Delft Institute of Applied Mathematics, TU Delft Matthias Möller 19 November 2014 M. Möller Time stepping / 54
2 Part I Space-Time Discretization M. Möller (DIAM@TUDelft) Time stepping / 54
3 Space-Time discretization Initial-boundary value problem (IBVP) u(x,t) t + Lu(x, t) = f in Ω (0, T ) time-dependent PDE u(x, t) = g 0 on Γ 0 (0, T ) n u(x, t) = g 1 on Γ 1 (0, T ) boundary conditions n u(x, t) + αu(x) = g 2 on Γ 2 (0, T ) u(x, 0) = u 0 in Ω at t = 0 initial condition Observation: transient term u t represents linear convection with unit velocity along the t-direction ( forward in time ) Interpret n-dimensional IBVP as n + 1-dimensional space-time BVP Use FEM-space discretization technique to approximate the n + 1-dimensional BVP by a set of algebraic equations M. Möller (DIAM@TUDelft) Time stepping / 54
4 Example: Transient heat equation Experiment: rod with initial constant temperature is heated from the left Transient heat equation in 1D u u = 0 t x 2 in (0, L) (0, T ) time-dependent PDE u(0, t) = 1 at x = 0, in (0, T ) Dirichlet boundary conditions u(1,t) = 0 x at x = L, in (0, T ) Neumann boundary conditions u(x, 0) = u 0 in (0, L), at t = 0 initial condition Space-time heat equation L := α 2 t x 2 Lu = 0 in Ω stationary PDE u = 1 on Γ 0 Dirichlet boundary conditions T Γ 0 Γ final Ω Γ 1 u = 0 x on Γ1 Neumann boundary conditions u = u 0 on Γ init Dirichlet boundary conditions 0 0 Γ init L M. Möller (DIAM@TUDelft) Time stepping / 54
5 Example: Transient heat equation Transient heat equation Space-time heat equation 2.0 sec sec 1.0 sec u sec 0.0 sec t x Grid: x = 0.05, t = 0.01 Attains steady-state limit u 1, that is, u t 0 for time t M. Möller (DIAM@TUDelft) Time stepping / 54
6 Take home lessons #1 Time-dependent IBVPs in nd can be interpreted as stationary BVPs in (n + 1)D subject to additional boundary conditions at Γ init Solution values for all time levels can be computed simultaneously Space-time discretization can become computationally expensive (large memory requirements and computing times especially in 3D) Final time T must be known a priori and cannot be changed Mesh generation in 4D is non-trivial (need for special elements) Solutions to boundary value problems can be computed by marching the solution of the associated transient problem to the steady state M. Möller (DIAM@TUDelft) Time stepping / 54
7 Part II Single-Step Methods M. Möller Time stepping / 54
8 Time-stepping techniques Idea: Unsteady flows are parabolic in time use time-stepping methods to advance transient solutions step-by-step in time time future present past zone of influence domain of dependence space Time discretization: 0 = t 0 < t 1 < t 2 < t M = T and solve IBVP on short time intervals (t n, t n+1 ), where t n+1 = t n + t t 0 t 1 t 2 t n t n+1 T Initialize solution values by the initial condition u 0 u 0 in Ω Given u n u(t n ) use it as initial condition to compute u n+1 u(t n+1 ) M. Möller (DIAM@TUDelft) Time stepping / 54
9 Time-stepping techniques Time-dependent PDE u(x, t) t + Lu(x, t) = f in Ω (0, T ) Observation: The space and time variables x and t are essentially decoupled and can be discretized independently Space discretization by the FEM yields time-dependent unknowns u h (t) Method of lines (MOL): du h (t) dt L L h yields ODE system for u i (t) + L h u h (t) = f h on (t n, t n+1 ) semi-discretized equations Remark: It is also possible to perform the time discretization before the discretization in space (later in this lecture) M. Möller (DIAM@TUDelft) Time stepping / 54
10 Method of lines (MOL) MOL du h(t) dt + L h u h (t) = f h, R(t, u h (t)) = f h L h u h (t) Initial value problem { duh (t) dt u h (t n ) = u n h = R(t, u h (t)) on (t n, t n+1 ) Exact integration t n+1 t n du h dt dt = un+1 h u n h = t n+1 R(t, u t n h (t)) dt Mean value theorem: τ (t n, t n+1 ) u n+1 R(τ, u h (τ)) h = u n h + R(τ, u h(τ)) t, t = t n+1 t n t n τ t n+1 Idea: evaluate the integral numerically using a suitable quadrature rule M. Möller (DIAM@TUDelft) Time stepping / 54
11 Two-level time-stepping schemes Numerical integration on the time interval (t n, t n+1 ) R R R R t n left endpoint t n+1 t t n t n+1 t right endpoint t n t n+1 t trapezoidal rule t n t n+1 t midpoint rule Forward Euler u n+1 h = u n h + R(t n, u n h) t + O( t 2 ) Backward Euler u n+1 h = u n h + R(t n+1, u n+1 h ) t + O( t 2 ) [ Crank-Nicolson u n+1 h = u n h R(t n, u n h) + R(t n+1, u n+1 h ) ] t + O( t 3 ) Leapfrog method u n+1 h = u n h + R(t n+1/2, u n+1/2 h ) t + O( t 3 ) M. Möller (DIAM@TUDelft) Time stepping / 54
12 Two-level θ-time-stepping schemes Standard finite difference discretization of the time derivative u n+1 h u n h t = θr(t n+1, u n+1 h ) + (1 θ)r(t n, u n h), 0 θ 1 where 0 θ 1 is the implicitness parameter θ = 0 forward Euler O( t) explicit update of u n+1 h θ = 1/2 Crank-Nicolson O( t 2 ) solution of (linear) implicit θ = 1 backward Euler O( t) algebraic system For θ 1 2 the θ-scheme is A-stable ( numerical solution is bounded) So, what it the best time-stepping method then? It depends M. Möller (DIAM@TUDelft) Time stepping / 54
13 Properties of the θ-scheme The optimal choice of the time-stepping scheme depends on its purpose: to march the numerical solution to a steady state limit (intermediate results are immaterial) use large t to reach convergence quickly to obtain a time-accurate discretization of a highly dynamic flow problem (evolution details are essential and must be captured) use sufficiently small time steps t to reduce temporal errors or Time discretization t 0 t 1 t 2... t n = n t... T = M t Accumulation of truncation errors n = 0, 1,..., M 1, M = T t ɛ loc τ = O( t p ) ɛ glob τ = Mɛ loc τ = O( t p 1 ) The order of a time-stepping method (i.e., the asymptotic rate at which the error is reduced as t 0) is not the sole indicator of accuracy M. Möller (DIAM@TUDelft) Time stepping / 54
14 Explicit vs. implicit time-stepping schemes Explicit time-stepping schemes easy to implement and parallelize, low cost per time step a good starting point for the development of CFD software small time steps are required for stability reasons, especially if the velocity and/or mesh size are varying strongly extremely inefficient for solution of stationary problems unless local time-stepping i.e. t = t(x) is employed Remark: Use the forward Euler method (θ = 0) with care since it may be unconditionally unstable (e.g., for standard Galerkin in space) M. Möller (DIAM@TUDelft) Time stepping / 54
15 Explicit vs. implicit time-stepping schemes Implicit time-stepping schemes stable over a wide range of time steps, sometimes unconditionally constitute excellent iterative solvers for steady-state problems difficult to implement and parallelize, high cost per time step insufficiently accurate for truly transient problems at large t convergence of linear solvers deteriorates/fails as t increases M. Möller (DIAM@TUDelft) Time stepping / 54
16 Galerkin method of lines Continuous problem: find u V such that Ω [ ] u v t + Lu f dx = 0 v V, t (t n, t n+1 ) FEM approximation u h (x, t) = N u j (t)ϕ j (x), u n i u(x i, t n ) j=1 Semi-discrete problem: find u h V h such that N [ ] N du j ϕ iϕ j dx Ω dt + [ ] ϕ ilϕ j dx u j = Ω j=1 } {{ } m ij j=1 } {{ } a ij ϕ if dx, Ω } {{ } b i i = 1,..., N, t (t n, t n+1 ) Differential-Algebraic Equation (DAE) M C du dt + Au = b where M C = {m ij } is the mass matrix and u(t) = [u 1 (t),..., u N (t)] T M. Möller (DIAM@TUDelft) Time stepping / 54
17 Galerkin method of lines du Galerkin MOL M C dt + Au = b, 1 R(t, u) = M [b Au] C [M C θ ta]u n+1 = [M C + (1 θ) ta]u n + tb n+θ where b n+θ = θb n+1 + (1 θ)b n, 0 θ 1, n = 0, 1,..., M 1 Linear algebraic system needs to be solved even for explicit scheme (θ = 0) Idea: Replace M C by the lumped mass matrix M L = diag{m i } where m i = m ij = Ω ϕ i, ϕ j dx = Ω ϕ i dx since ϕ 1 j j j for most finite elements. 1D case FDM=FVM=FEM+mass lumping Lumped-Galerkin MOL du M L dt + Au = b, 1 R(t, u) = M [b Au] L M. Möller (DIAM@TUDelft) Time stepping / 54
18 Take home lessons #2 Due to the parabolic nature of unsteady flows, time-stepping schemes can be used to advance transient solutions step-by-step Space discretizations techniques yield a first-order ODE system which is integrated forward in time numerically (method of lines) Galerkin method of lines leads to differential-algebraic equations unless the consistent mass matrix is replaced by the lumped mass matrix The optimal choice of the time-stepping scheme depends on its purpose (steady-state vs. dynamic flows, explicit vs. implicit) M. Möller (DIAM@TUDelft) Time stepping / 54
19 Part III Multi-Step and other Advanced Methods M. Möller Time stepping / 54
20 Fractional-step θ-scheme Given parameters θ (0, 1), θ = 1 2θ, and α [0, 1] subdivide time interval (t n, t n+1 ) into three substeps and update the solution as follows Step 1. u n+θ = u n + [αr(t n+θ, u n+θ ) + (1 α)r(t n, u n )]θ t Step 2. u n+1 θ = u n+θ + [(1 α)r(t n+1 θ, u n+1 θ ) + αr(t n+θ, u n+θ )]θ Step 3. u n+1 = u n+1 θ + [αr(t n+1, u n+1 ) Properties of this time-stepping method + (1 α)r(t n+1 θ, u n+1 θ )]θ t second-order accurate in the special case θ = coefficient matrices are the same for all substeps if α = 1 2θ 1 θ combine the advantages of Crank-Nicolson and backward Euler M. Möller (DIAM@TUDelft) Time stepping / 54
21 Predictor-corrector and multipoint methods Objective: to combine the simplicity of explicit schemes and robustness of implicit ones in the framework of a fractional-step algorithm, e.g., 1. Predictor ũ n+1 = u n + R(t n, u n ) t forward Euler 2. Corrector u n+1 = u n [R(tn, u n ) Crank-Nicolson +R(t n+1, ũ n+1 )] t or u n+1 = u n + R(t n+1, ũ n+1 ) t backward Euler Stability still leaves a lot to be desired, additional correction steps usually do not pay off since iterations may diverge if t is too large Order barrier: two-level methods are at most second-order accurate, so extra points are needed to construct higher-order integration schemes t 0 t 1 t 2 t n 2 t n t n+1 t n 1 T M. Möller (DIAM@TUDelft) Time stepping / 54
22 Adams methods Derivation: polynomial fitting R past present future Truncation error: ɛ glob τ = O( t p ) for polynomials of degree p 1 which interpolate function values at p points Adams-Bashforth methods (explicit) p = 1 u n+1 = u n + tr(t n, u n ) forward Euler p = 2 p = 3 t n 2 t n 1 t n t n+1 t u n+1 = u n + t 2 [3R(tn, u n ) R(t n 1, u n 1 )] u n+1 = u n + t 12 [23R(tn, u n ) 16R(t n 1, u n 1 ) + 5R(t n 2, u n 2 )] Adams-Moulton methods (implicit) p = 1 u n+1 = u n + tr(t n+1, u n+1 ) backward Euler p = 2 p = 3 u n+1 = u n + t 2 [R(tn+1, u n+1 ) + R(t n, u n )] Crank-Nicolson u n+1 = u n + t 12 [5R(tn+1, u n+1 ) + 8R(t n, u n ) R(t n 1, u n 1 )] M. Möller (DIAM@TUDelft) Time stepping / 54
23 Adams methods Predictor-corrector algorithm 1 Compute ũ n+1 using an Adams-Bashforth method of order p 1 2 Compute u n+1 using an Adams-Moulton method of order p with predicted value R(t n+1, ũ n+1 ) instead of R(t n+1, u n+1 ) methods of any order are easy to derive and implement only one function evaluation per time step is performed error estimators for ODEs can be used to adapt the order other methods are needed to start/restart the calculation time step is difficult to change (coefficients are different) tend to be unstable and produce nonphysical oscillations M. Möller (DIAM@TUDelft) Time stepping / 54
24 Runge-Kutta methods Multipredictor-multicorrector algorithms of order p p = 2 ũ n+1/2 = u n + t 2 R(tn, u n ) forward Euler / predictor u n+1 = u n + tr(t n+1/2, ũ n+1/2 ) midpoint rule / corrector p = 4 ũ n+1/2 = u n + t 2 R(tn, u n ) forward Euler / predictor û n+1/2 = u n + t 2 R(tn+1/2, ũ n+1/2 ) backward Euler / corrector ũ n+1 = u n + tr(t n+1/2, û n+1/2 ) midpoint rule / predictor u n+1 = u n + t 6 [R(tn, u n ) + 2R(t n+1/2, ũ n+1/2 ) Simpson rule + 2R(t n+1/2, û n+1/2 ) + R(t n+1, ũ n+1 )] corrector There exist embedded Runge-Kutta methods which perform extra steps in order to estimate the error and adjust t in an adaptive fashion M. Möller (DIAM@TUDelft) Time stepping / 54
25 Runge-Kutta methods Runge-Kutta methods self-starting, easy to operate with variable time steps more stable and accurate than Adams methods of the same order high order approximations are rather difficult to derive; p function evaluations per time step are required more expensive than Adams methods of comparable order Use of variable time step sizes makes it possible to achieve the desired accuracy at a relatively low computational cost Explicit method: use the largest time step satisfying the stability condition Implicit method: estimate the error and adjust the time step if necessary M. Möller (DIAM@TUDelft) Time stepping / 54
26 Take home lessons #3 Advantages of Crank-Nicolson and backward Euler method can be combined by using the fractional-step θ-scheme Second-order barrier of two-level methods can be avoided by adding extra points from previous time steps (Adams methods) Simple time-stepping schemes can be combined in a predictor-corrector algorithm (Runge-Kutta methods) M. Möller (DIAM@TUDelft) Time stepping / 54
27 Part IV Analysis of numerical dissipation and dispersion M. Möller Time stepping / 54
28 Properties of numerical methods Example: convection-dominated / hyperbolic PDEs P e 1, Re 1 u t + v u x = 0, v > 0 1 High-order numerical method wiggles (phase errors) Low-order numerical method smearing (amplitude errors) Objective: to analyze the creation of wiggles and the smearing of solutions; design accurate methods which do not lead to spurious oscillations M. Möller (DIAM@TUDelft) Time stepping / 54
29 Modified equation method Motivation: PDEs are difficult or impossible to solve analytically but their qualitative behavior is easier to predict than that of discretized equations Observation: exact solution (no roundoff errors) of the discretized equations satisfies a PDE which is generally different from the one to be solved Original PDE u t + Lu = 0 Numerical method solves modified equation u t + Lu = 2p u α 2p x + 2p+1 u α 2p 2p+1 x 2p+1 p=1 p=1 1 Expand all nodal values in the difference scheme in a double Taylor series about a single point (x i, t n ) of the space-time mesh PDE 2 Express high-order time derivatives and mixed derivatives in this PDE in terms of space derivatives only modified equation M. Möller (DIAM@TUDelft) Time stepping / 54
30 Derivation of the modified equation Example: pure convection equation u t + v u x = 0, v > 0 discretized by BDS in space FE in time u n+1 i u n i t + v un i un i 1 x = 0 (upwind) ( ) Taylor series expansions about the point (x i, t n ) u n+1 i = u n i + t ( ) u n + ( t)2 t i 2 u n i 1 = u n i x ( ) u n + ( x)2 x i 2 Substitution into the difference scheme yields ( u + v u t x ) ( ) n = t 2 n u i 2 t 2 i ( ) 2 n u + ( t)3 t 2 6 i ( ) 2 n u ( x)3 x 2 6 i ( ) 3 n u +... t 3 i ( ) 3 n u +... x 3 i ( ) ( t)2 3 n ( ) u + v x 2 n ( ) u v( x)2 3 n u t 3 2 x i 2 6 x i 3 i Next step: Replace high-order time derivatives by space derivatives M. Möller (DIAM@TUDelft) Time stepping / 54
31 Derivation of the modified equation (1) 2 u t 2 Differentiate ( ) with respect to t to obtain + v 2 u x t = t 2 3 u t 3 ( t)2 6 4 u t 4 + v x 2 3 u x 2 t v( x)2 6 4 u x 3 t +... Differentiate ( ) with respect to x and multiply by v to obtain (2) v 2 u t x + v2 2 u x 2 = v t 2 3 u t 2 x v( t)2 4 u 6 t 3 x + v2 x 3 u 2 x v2 ( x) u x (3) 2 u t 2 Subtract (2) from (1) and drop high-order terms to obtain [ ] [ ] = v 2 2 u + t 3 u + v 3 u + O( t) + x v 3 u x 2 2 t 3 t 2 x 2 x 2 t v2 3 u + O( x) x 3 Replace 3rd order time derivative and mixed derivatives by space derivatives M. Möller (DIAM@TUDelft) Time stepping / 54
32 Derivation of the modified equation Differentiate (2) with respect to x and divide by v to obtain (4) 3 u x 2 t = v 3 u x + O[ t, x] 3 Differentiate (3) with respect to t and x to obtain respectively (5) 3 u t = v 2 3 u 3 x 2 t + O[ t, x] and (6) 3 u t 2 x = v2 3 u x + O[ t, x] 3 Equations (5) and (4) imply (7) 3 u t 3 = v 3 3 u x + O[ t, x] 3 (8) 2 u t 2 Plugging equations (4), (6), and (7) into (3) yields = v 2 2 u x + v 2 (v t x) 3 u 2 x + O[ t, x] 3 M. Möller (DIAM@TUDelft) Time stepping / 54
33 Derivation of the modified equation Substitute (7) and (8) into ( ) to obtain the modified equation [ ] u + v u = v2 t 2 u + (v t x) 3 u + v3 ( t) 2 3 u + v x 2 u v( x)2 3 u +... t x 2 x 2 x 3 6 x 3 2 x 2 6 x 3 Reformulation in terms of the Courant number ν = v t x yields Modified equation for FE/BDS discretization of u + v u = 0, v > 0 t x u t + v u = v x x 2 (1 u ν) 2 x 2 + v( x)2 6 }{{} original PDE }{{} numerical dissipation (3ν 2ν 2 1) 3 u x 3 } {{ } numerical dispersion +... Remark: the CFL stability condition ν 1 must be satisfied for the discrete problem to be well-posed. In the case ν > 1, the numerical diffusion coefficient v x 2 (1 ν) is negative, which corresponds to a backward heat equation. M. Möller (DIAM@TUDelft) Time stepping / 54
34 Significance of terms in the modified equation Exact solution of the discretized equation satisfies the modified equation u t + Lu = 2p u α 2p x + 2p+1 u α 2p 2p+1, with L = v x2p+1 x p=1 p=1 Even-order derivatives 2p u x 2p cause numerical dissipation smearing (amplitude errors) Odd-order derivatives 2p+1 u x 2p+1 cause numerical dispersion wiggles (phase errors) Qualitative analysis: the numerical behavior of the discretization scheme largely depends on the relative importance of dispersive and dissipative effects M. Möller (DIAM@TUDelft) Time stepping / 54
35 Forward Euler central difference scheme Stability condition (necessary but not sufficient) Coefficients of the even-order derivatives in the modified equation must have alternating signs, the one for the second-order term being positive Modified equation for the FE/CDS scheme u t + v u x = v x 2 ν 2 u where ν = v t x x 2 v( x)2 6 is the Courant number (1 + 2ν 2 ) 3 u x FE/CDS scheme is unconditionally unstable since the coefficient v x 2 ν = t v2 < 0 v 0 2 of the second-order term is negative for nonzero velocity M. Möller (DIAM@TUDelft) Time stepping / 54
36 Forward Euler finite element method Stability condition (necessary but not sufficient) Coefficients of the even-order derivatives in the modified equation must have alternating signs, the one for the second-order term being positive Modified equation for the FE/FEM scheme with ν = v t x u t + v u x = v x 2 ν 2 u x 2 v( x)2 3 ν 2 3 u x Elimination of leading dispersion error due to space discretization FE/FEM scheme is unconditionally unstable since the coefficient v x 2 ν = t v2 < 0 v 0 2 of the second-order term is negative for nonzero velocity M. Möller (DIAM@TUDelft) Time stepping / 54
37 Stabilization by means of artificial diffusion Stabilization strategy: if the necessary stability condition is violated, it can be enforced by adding artificial diffusion to the numerical scheme: Stabilized methods +δ(v ) 2 u Nonoscillatory methods +δ(v ) 2 u + ɛ(u) u streamline diffusion shock-capturing viscosity Remark: in the one-dimensional case both terms are proportional to 2 u x 2 Free parameters depending on the mesh size h and the residual R(u) δ = c δh 1+ v, ɛ(u) = c ɛh 2 R(u) Problem: how to determine proper values of the constants c δ and c ɛ??? M. Möller (DIAM@TUDelft) Time stepping / 54
38 Take home lessons #4 The exact solution of the discretized equations does not satisfy the original PDE but the modified equation Even-order derivatives in the modified equation cause numerical dissipation which affects to amplitude errors ( smearing) Odd-order derivatives in the modified equation cause numerical dispersion which affects the phase errors ( wiggles) For stability, the coefficients of the even-order derivatives must have alternating sign, the one for the second-order term being positive Numerical schemes can be stabilized by adding artificial diffusion M. Möller (DIAM@TUDelft) Time stepping / 54
39 Part V Lax-Wendroff Time-Stepping Method M. Möller (DIAM@TUDelft) Time stepping / 54
40 Lax-Wendroff method Objective: to use a high-order time-stepping method for stabilization Consider a time-dependent PDE u t + Lu = 0 in Ω (0, T ) 1 Discretize it in time by means of the Taylor series expansion u n+1 = u n + t ( ) u n ( ) t + ( t) 2 n 2 u 2 t + O( t 3 ) 2 2 Transform time derivatives into space derivatives using the PDE u t = Lu, 2 u t 2 = t ( u t ) = u t ( Lu) = L t = L2 u 3 Substitute the resulting expressions into the Taylor series u n+1 = u n tlu n + ( t)2 2 L 2 u n + O( t 3 ) 4 Perform space discretization using the FDM / FVM / FEM M. Möller (DIAM@TUDelft) Time stepping / 54
41 Lax-Wendroff central difference scheme Example: pure convection equation Time derivatives Semi-discrete scheme u t + v u x = 0 L = v x u u t = v x, 2 u t 2 u n+1 = u n v t ( ) u n+ v 2 ( t) 2 x 2 = v 2 2 u x 2 ( 2 u x 2 ) n+ O( t 3 ) Central difference approximation (CDS) in space ( u ) ( x i = ui+1 ui 1 2 x + O( x 2 ), 2 u x = )i ui+1 2ui+ui 1 2 ( x) + O( x 2 ) 2 Fully discrete LW/CDS scheme u n i + v un i+1 u n i 1 } t {{ 2 x } FE/CDS scheme u n+1 i = v2 t 2 u n i+1 2u n i + u n i 1 ( x) 2 } {{ } numerical dissipation +O[ t 2, x 2 ] FE/CDS scheme stabilized by artificial diffusion (no adjustable parameter) M. Möller (DIAM@TUDelft) Time stepping / 54
42 Lax-Wendroff central difference scheme Modified equation for the LW/CDS scheme u t + v u x = v( x)2 (1 ν 2 ) 3 u 6 x 3 v( x)3 ν(1 ν 2 ) 4 u 8 x 4 v( x)4 120 (1 + 5ν2 6ν 4 ) 5 u x , ν = v t x conditionally stable for ν 2 1 in 1D, ν in 2D, ν in 3D the second-order derivative (leading dissipation error) has been eliminated the leading truncation error vanishes for ν 2 = 1 (unit CFL property) the negative dispersion coefficient corresponds to a lagging phase error i. e. harmonics travel too slow, spurious oscillations occur behind steep fronts M. Möller (DIAM@TUDelft) Time stepping / 54
43 Lax-Wendroff finite element method Modified equation for the LW/FEM scheme u t + v u x =v( x)2 ν 2 3 u 6 x 3 v( x)3 ν(1 3ν 2 ) 4 u 24 x 4 + v( x)4 15 ( ν2 + 9ν 4 ) 5 u x , ν = v t x conditionally stable for ν in 1D, ν in 2D, ν in 3D the second-order derivative (leading dissipation error) has been eliminated the truncation error does not vanish for ν 2 = 1 (no unit CFL property) the positive dispersion coefficient corresponds to a leading phase error i. e. harmonics travel too fast, spurious oscillations occur ahead of steep fronts M. Möller (DIAM@TUDelft) Time stepping / 54
44 Part VI Taylor-Galerkin Time-Stepping Methods M. Möller Time stepping / 54
45 Taylor-Galerkin methods TG methods: family of high-order time-stepping schemes which stabilize the convective terms by means of intrinsic streamline diffusion (Donea 84) Generic procedure (to be adapted for a particular scheme) 1 Taylor series expansion(s) in time up to the third/fourth order u n+1 = u n +..., u n = u n+1... or u n 1 = u n... 2 Transform time derivatives into space derivatives avoiding thirdorder space derivatives. As example, let u t + Lu = 0 then u t = Lu, 2 u t = L 2 u, 3 u 2 t = L 2 u 3 t = L2 un+1 u n t + O( t) 3 Substitute the resulting expressions into the Taylor series and perform space discretization using the FEM / FDM / FVM M. Möller (DIAM@TUDelft) Time stepping / 54
46 Euler Taylor-Galerkin scheme Consider convection-dominated PDE u t + Lu = 0 in Ω (0, T ) Taylor series expansion up to the third order u n+1 = u n + t ( ) u n ( ) t + ( t) 2 n ( ) 2 u 2 t + ( t) 3 n 3 u 2 6 t + O( t 3 ) 3 Substitution of u t = Lu, 2 u t 2 = L 2 u, 3 u t 3 yields the semi-discrete FE/TG scheme = L 2 un+1 u n t + O( t) u n+1 = u n tlu n + ( t)2 2 L 2 u n + ( t)2 6 L 2 (u n+1 u n ) + O( t 4 ) The Lax-Wendroff scheme is recovered for u n+1 = u n (steady state) Equivalent form ] [I ( t)2 6 L 2 u n+1 u n t = Lu n + t 2 L2 u n M. Möller (DIAM@TUDelft) Time stepping / 54
47 Euler Taylor-Galerkin scheme Example: pure convection equation u t + v u x = 0, L = v x discretized in space by the Galerkin FEM with linear elements Modified equation for the FE/TG scheme u t + v u x = v( x)3 24 ν(1 ν 2 ) 4 u x 4 + v( x)4 180 (1 5ν2 + 4ν 4 ) 5 u x ν = v t x conditionally stable for ν 2 1 in 1D, ν in 2D, ν in 3D the leading truncation error vanishes for ν 2 = 1 (unit CFL property) the leading dispersion error is of higher order than that for LW/FEM M. Möller (DIAM@TUDelft) Time stepping / 54
48 Take home lessons #5 Numerical methods can be stabilized using high-order time-stepping schemes which introduce artificial diffusion (no free parameter) Such methods can be constructed by the three-step approach: discretize the PDE in time by a Taylor series expansion replace time derivatives by space derivatives using the PDE perform discretization in space by FDM / FVM / FEM Lax-Wendroff methods involve space derivative which may be significantly higher than in the original PDE (e.g. 3 u t 3 = L 3 u) Taylor-Galerkin methods avoid higher-order space derivatives but their derivation needs to be adapted for each particular scheme M. Möller (DIAM@TUDelft) Time stepping / 54
49 Part VII Adaptive Time-Step Control M. Möller Time stepping / 54
50 Automatic time step control Objective: to make sure that u u t T OL (prescribed tolerance) Assume that the error at t = t n is equal to zero (no accumulation) Local truncation error u t = u + t 2 e(u) + O( t 4 ) u m t = u + m 2 t 2 e(u) + O( t 4 ) Heuristic error analysis e(u) u m t u t t 2 (m 2 1) Estimate of the relative error u u t ( ) 2 t u m t u t t m 2 = T OL 1 Adaptive time stepping t 2 = T OL t2 (m 2 1) u t u m t Richardson extrapolation u = m2 u t u m t m O( t 4 ) M. Möller (DIAM@TUDelft) Time stepping / 54
51 Practical implementation Given the old solution u n do: 1 Make one large time step of size m t to compute u m t 2 Make m small time steps each of size t to compute u t 3 Evaluate the relative solution changes u t u m t 4 Calculate the optimal value t for the next time step 5 If t t, reset the solution and go back to the beginning 6 Set u n+1 = u t or perform Richardson extrapolation enhance robustness and overall efficiency of the code simulation results are more credible due to estimated error cost per time step increases substantially (u m t may be as expensive to obtain as u t due to slow convergence at large time steps) M. Möller (DIAM@TUDelft) Time stepping / 54
52 Evolutionary PID controller Idea: Monitor the relative changes e n = un+1 u n u n of an indicator variable u and keep them close to some prescribed tolerance T OL Strategy: Use proportional-integral-differential (PID) controller M. Möller (DIAM@TUDelft) Time stepping / 54
53 Evolutionary PID controller Given the old solution u n do: 1 Make one step of size t to compute u n+1 2 Compute the relative changes e n = un+1 u n u n 3 If e n < ɛ reject solution and repeat time step using t = ɛ e n t n 4 Adjust time step smoothly to approach the prescribed tolerance ( ) kp ( ) ki ( en 1 T OL e 2 ) kd t n+1 = n 1 t n e n e n e n e n 2 5 Limit the growth and reduction of the time step so that t min t n+1 t max, l t n+1 t n L Empirical PID parameter k P = 0.075, k I = 0.175, k D = 0.01 M. Möller (DIAM@TUDelft) Time stepping / 54
54 Take home lessons #6 Adaptive time-stepping enhances the robustness of the code, the overall efficiency and the credibility of simulation results Automatic time step control is based on local truncation error analysis PID controllers provide an easy to implement and cost effective mechanism to keep the relative changes close to a prescribed tolerance M. Möller (DIAM@TUDelft) Time stepping / 54
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