7 Hyperbolic Differential Equations

Transcription

1 Numerical Analysis of Differential Equations Hyperbolic Differential Equations While parabolic equations model diffusion processes, hyperbolic equations model wave propagation and transport phenomena. Important application areas include Acoustics Electromagnetics Seismics Optics Fluid Mechanics Modeling the propagation of small perturbations of an equilibrium typically leads to linear equations. 7 Hyperbolic Differential Equations TU Bergakademie Freiberg, SS 2012

2 Numerical Analysis of Differential Equations The Linear Advection Equation We consider the simplest hyperbolic PDE, the linear advection equation, sometimes called the one-way wave equation. The Cauchy problem for this PDE reads with a constant c. u t + cu x = 0, x R, t > 0, (7.1a) u(x, 0) = u 0 (x), (7.1b) The solution of (7.1) poses little difficulty, as the exact solution is easily verified to be u(x, t) = u 0 (x ct). Nonetheless, this simple problem serves to illustrate typical difficulties encountered in the numerical solution of hyperbolic PDEs. 7.1 The Linear Advection Equation TU Bergakademie Freiberg, SS 2012

3 Numerical Analysis of Differential Equations 245 We again introduce a space-time grid {(x j, t n ) : x j = j x, j Z; t n = n t, n N 0 }. Natural explicit scheme: central difference approximation of u x combined with explicit Euler method in time, results in or, in explicit form, U n+1 j t U n j + c U n j+1 U n j 1 2 x = 0 U n+1 j = U n j c t 2 x (U n j+1 U n j 1). (7.2) As we will later see, this is a useless method. However, replacing Uj n by the average 1 2 (U j 1 n +U j+1 n ) leads to the eminently useful Lax-Friedrichs method U n+1 j = 1 2 (U n j 1 + U n j+1) c t 2 x (U n j+1 U n j 1). (7.3) 7.1 The Linear Advection Equation TU Bergakademie Freiberg, SS 2012

4 Numerical Analysis of Differential Equations 246 We will see that (7.3) is Lax-Richtmyer stable if c t x 1. In particular, this permits choosing a timestep of t = O( x). 7.1 The Linear Advection Equation TU Bergakademie Freiberg, SS 2012

5 Numerical Analysis of Differential Equations Method of Lines Discretisation Formulating an IBVP for (7.1a) on the spatial domain x (0, 1) requires a boundary condition on one of the two endpoints. When c > 0, a boundary condition at x = 0 leads to a well-posed problem, e.g. u(0, t) = g 0 (t), t 0. In this case the left boundary is called the inflow boundary, and the right the outflow boundary. Their roles are reversed when c < 0. For the two schemes considered so far, some numerical boundary condition must be chosen to determine the values at the outflow boundary, which complicates the stability analysis. We therefore simplify matters by choosing periodic boundary conditions and obtain the IBVP... u(0, t) = u(1, t), t 0, 7.2 Method of Lines Discretisation TU Bergakademie Freiberg, SS 2012

6 Numerical Analysis of Differential Equations 248 u t + cu x = 0, x (0, 1), t > 0, (7.4a) u(0, t) = u(1, t), t 0, (7.4b) u(x, 0) = u 0 (x). The vector of approximations at the spatial gridpoints (semidiscretization!) U 1 (t) U (t) =., U j(t) u(x j, t), U J+1 (t) must then solve the system of ODEs U 1(t) = c [ U2 (t) U J+1 (t) ], 2 x U j(t) = c [ Uj+1 (t) U j 1 (t) ], 2 j J, 2 x U J+1(t) = c [ U1 (t) U J (t) ]. 2 x (7.4c) 7.2 Method of Lines Discretisation TU Bergakademie Freiberg, SS 2012

7 Numerical Analysis of Differential Equations 249 In matrix notation: U (t) = AU (t), U (0) = U 0, where [U 0 ] j = u 0 (x j ), and A = c 2 x R (J+1) (J+1). (7.5) The matrix A is skew symmetric (A = A), and its eigenvalues are λ j = ic x with associated eigenvectors sin(2πj x), j = 1, 2,..., J + 1, [v j ] k = e 2πikj x, j = 1, 2,..., J Method of Lines Discretisation TU Bergakademie Freiberg, SS 2012

8 Numerical Analysis of Differential Equations 250 The eigenvalues of A are thus contained in the interval [ ic x, ic x] on the imaginary axis. For absolute stability in the method of lines, a time-stepping scheme is required whose region of absolute stability contains this interval. For the explicit Euler method (7.2) this is not the case for any value of t/ x. This method is therefore not absolutely stable for any value of this grid ratio. A different choice, however, satisfies the weaker variant (6.11) of Lax- Richtmyer stability: for the eigenvalues of the matrix B( t) = I + ta we have 1 + tλ j 2 = 1 + ( ) 2 c t sin 2 (2πj x) 1 + x ( ) 2 c t, x so that choosing t = x 2 results in 1 + tλ j c 2 t and thus to the bound B n = (I + ta) n (1 + c 2 t) n/2 e c2 T/2, n t T. 7.2 Method of Lines Discretisation TU Bergakademie Freiberg, SS 2012

9 Numerical Analysis of Differential Equations The Leapfrog Method If, in (7.1a) or (7.4a), using the same central difference formlula for u x, one uses the midpoint rule y n+1 = y n t f(t n, y n ) in place of the explicit Euler method, one obtains the so-called leapfrog scheme U n+1 j = U n 1 j c t x (U j+1 n Uj 1). n (7.6) Since the region of absolute stability of the midpoint rule is the interval {iy : 1 < y < 1} on the imaginary axis, the leapfrog scheme is absolutely stable if c t x < The Leapfrog Method TU Bergakademie Freiberg, SS 2012

10 Numerical Analysis of Differential Equations The Lax-Friedrichs Scheme The replacement 1 2 (U n j 1 + U n j+1 ) = U n j (U n j 1 2U n j + U n j+1 ) modifies the Lax-Friedrichs scheme (7.3) to U n+1 j = U n j c t 2 x (U n j+1 U n j 1) (U n j 1 2U n j and, upon rearranging the terms to + U n j+1), U n+1 j t U n j + c U n j+1 U n j 1 2 x = x2 2 t U n j 1 2U n j + U n j+1 x 2. (7.7) Consistency analysis reveals this scheme to be consistent with the PDE u t + cu x = 0. However, it looks more like a discretization of the advectiondiffusion equation u t + cu x = ɛu xx, ɛ = x2 2 t. 7.4 The Lax-Friedrichs Scheme TU Bergakademie Freiberg, SS 2012

11 Numerical Analysis of Differential Equations 253 The scheme (7.7) thus consists of the explicit Euler method applied to A ɛ = c 2 x U (t) = A ɛ U (t), with (7.8) ɛ x The matrix (7.8) is obtained from the matrix (7.5) by adding a small multiple of a symmetric, negative definite difference operator, causing its eigelvalues to move away from the imaginary axis into the left half plane. Due to the periodic boundary conditions both matrices are circulant and therefore have the same (orthogonal) eigenvectors The eigenvalues are λ ɛ j = ic x sin(2πj x) 2ɛ (1 cos(2πj x), j = 1, 2,..., J. (7.9) x2 7.4 The Lax-Friedrichs Scheme TU Bergakademie Freiberg, SS 2012

12 Numerical Analysis of Differential Equations Eigenvalues of ta ɛ for c = 1, x = 1/50 t = 0.8 x for different values of ɛ expl. Euler (ɛ = 0) ɛ = ɛ = Lax-Friedrichs (ɛ = ) 7.4 The Lax-Friedrichs Scheme TU Bergakademie Freiberg, SS 2012

13 Numerical Analysis of Differential Equations The Lax-Wendroff Scheme Possibilities for increasing consistency order w.r.t. t: Use 2nd order time stepping method such as midpoint rule, leapfrog. Disadvantages: involves three consecutive time levels (large memory requirements, especially in 3D), BC more difficult to implement. Leapfrog nondissipative (neutral stability), problems with variable coefficients/nonlinear terms Trapezoidal rule, but this is implicit. Two-stage Runge-Kutta method Taylor series approach applied to u (t) = Au(t), taking account of u (t) = Au (t) = A 2 u(t): u n+1 = u n + tau n + t2 2 A2 u n. 7.5 The Lax-Wendroff Scheme TU Bergakademie Freiberg, SS 2012

14 Numerical Analysis of Differential Equations 256 If A is the matrix from (7.5), this method written out is U n+1 j = U n j c t 2 x (U n j+1 U n j 1) + c2 t 2 8 x 2 (U n j 2 2U n j Bad: 5-point stencil in x, difficult near boundary. + U n j+2). Remedy: note that last term represents approximation to 1 2 c2 t 2 u xx. Replace this term by usual 3-point formula for u xx. This results in the Lax- Wendroff scheme U n+1 j = U n j c t 2 x (U n j+1 U n j 1) + c2 t 2 2 x 2 (U n j 1 2U n j + U n j+1). (7.10) 7.5 The Lax-Wendroff Scheme TU Bergakademie Freiberg, SS 2012

15 Numerical Analysis of Differential Equations Asymptotic Stability Note: The Lax-Wendroff scheme corresponds to the explicit Euler method applied to U (t) = A ɛ U (t) with A ɛ as in (7.8), except that now ɛ = c 2 t/2 in place of ɛ = x 2 /(2 t) as for the Lax-Friedrichs method. Just as in (7.9), we investigate the eigenvalues of ta ɛ : tλ ɛ j = i c t x sin(jπ x) + ( ) 2 c t (cos(jπ x 1), j = 1, 2,..., J. x These are located on the boundary of an ellipse with midpoint (c t/ x) 2 and semi-axes of length (c c/ x) 2 and c t/ x. This ellipse is contained in the region of absolute stability of the Euler scheme if c t x The Lax-Wendroff Scheme TU Bergakademie Freiberg, SS 2012

16 Numerical Analysis of Differential Equations 258 Based on the same parameters as in the previous section this figure shows the eigenvalues of the Lax-Wendroff method Lax-Wendroff (ɛ = 0.005) 7.5 The Lax-Wendroff Scheme TU Bergakademie Freiberg, SS 2012

17 Numerical Analysis of Differential Equations Upwind Schemes Besides the central difference formulas we have used up to now for approximating u x in the linear advection equation (7.1a), one could also consider using the one-sided difference formulas u x (x j, t) U j U j 1 x or u x (x j, t) U j+1 U j x Combined with a forward difference in time these lead to the schemes U n+1 j = U n j c t x (U j U j 1 ), U n+1 j = U n j c t x (U j+1 U j ). Both are consistent with order one in x and t.. (7.11a) (7.11b) 7.6 Upwind Schemes TU Bergakademie Freiberg, SS 2012

18 Numerical Analysis of Differential Equations 260 Despite their low order of accuracy these two schemes mimic a qualitative property of the solution of (7.1a), namely the propagation of information at speed c from left to right (c > 0) or right to left (c < 0). More precisely, for the exact solution we have u(x j, t + t) = u 0 (x j c(t + t)) = u(x j c t, t), i.e., the value of the solution in the next time step is determined by data which, at the current time step, lies to the left of x j (c > 0) or to the right of x j (c < 0). This suggests using (7.11a) when c > 0 and (7.11b) when c < Upwind Schemes TU Bergakademie Freiberg, SS 2012

19 Numerical Analysis of Differential Equations 261 To determine the stability properties of these schemes, note that (7.11a) may also be rearranged as U n+1 j = U n j c t 2 x (U n j+1 U n j 1) + c t 2 x (U n j+1 2U n j + U n j 1) (7.12) which is a scheme of the form (7.8) with ɛ = c x/2. Already established: such a scheme is stable if c t x 1 and 2 < 2ɛ t x 2 < Upwind Schemes TU Bergakademie Freiberg, SS 2012

20 Numerical Analysis of Differential Equations Numerical Example We consider the IVP (7.11a) with c = 1 and initial data u 0 (x) = e 20(x 2)2 + e (x 5)2 in the time range t [0, 17] and compare the exact solution with the approximations provided by the upwind, Lax-Wendroff and leapfrog schemes at the end of the time interval. The initial data consists of two Gaussian peaks, one of which is less well resolved by the spatial grid than the other. The spatial mesh size is x = 0.05, the timestep t = 0.8 x; we use homogeneous Dirichlet conditions as both (true and numerical) BC. The solid black line shows the extact solution, the red dots their numerical approximations at the grid points. 7.7 Numerical Example TU Bergakademie Freiberg, SS 2012

21 Numerical Analysis of Differential Equations Exakte Lösung t=0 1.5 Upwind, t= Numerical Example TU Bergakademie Freiberg, SS 2012

22 Numerical Analysis of Differential Equations Lax Wendroff, t= Leapfrog, t= Numerical Example TU Bergakademie Freiberg, SS 2012

23 Numerical Analysis of Differential Equations 265 Observations: The upwind scheme is strongly diffusive and approximates both peaks less well than the remaining methods. This is partly due to its low order of accuracy and can be improved by refining the mesh. Lax-Wendroff and leapfrog display a strong dispersion, i.e., different wave components in the numerical approximation are propagated with different speeds. 7.7 Numerical Example TU Bergakademie Freiberg, SS 2012

24 Numerical Analysis of Differential Equations 266 Goals of Section 7. You should be able to state the linear advection equation and discuss its solution properties; be familiar with the construction principles of the leapfrog, Lax-Friedrichs, Lax-Wendroff and upwind schemes; know the difference between absolute stability and Lax-Richtmyer stability; be able to carry out a von Neumann stability analysis for numerical schemes approximating the linear advection equation; be able to implement simple difference scheme for the linear advection equation. 7.7 Numerical Example TU Bergakademie Freiberg, SS 2012