Stability of Difference Schemes
The most powerful and most general method for constructing approximate solutions of hyperbolic partial differential equations with prescribed initial values is to discretize the space and time variables and solve the resulting finite system of equations.
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Abstract The most powerful and most general method for constructing approximate solutions of hyperbolic partial differential equations with prescribed initial values is to discretize the space and time variables and solve the resulting finite system of equations. How to discretize is a subtle matter, as we shall demonstrate. In this report, some of the proofs are only sketched; details can be found in Chap. 8 of my monograph “Hyperbolic Partial Differential Equations”, 2006, AMS. Keywords Hyperbolic PDE’s · Finite difference schemes · Convergence · Stability
One of the seminal observations of the Courant–Friedrichs–Lewy paper of 1928 was that in order for solutions of a difference equation to converge to the solution of the partial differential equation the difference scheme must use all the information contained in the initial data that influence the solution. To satisfy this condition, the ratio of the spatial discretization to the time discretization must be at least as large as the largest velocity with which signals propagate in solutions of the partial differential equation. This inequality is called the CFL condition. I well remember from the early days of computing, when physicists and engineers first undertook to solve numerically initial value problems, their utter astonishment to see the numerical solution blow up because they have unwittingly violated the CFL condition. The CFL condition is only a necessary condition for the convergence of difference schemes. Here is an example: discretize the scalar equation ut + ux = 0 by replacing the time derivative with a forward difference, and the space derivative with a symmetric difference. This scheme diverges, no matter how small the time discretization is compared to the space discretization. P.D. Lax () Courant Institute of Mathematical Sciences, New York University, New York, USA e-mail: [email protected] C.A. de Moura, C.S. Kubrusly (eds.), The Courant–Friedrichs–Lewy (CFL) Condition, DOI 10.1007/978-0-8176-8394-8_1, © Springer Science+Business Media New York 2013
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P.D. Lax
In this talk, I will report on sufficient conditions for the convergence of various difference schemes. I shall discuss a class of equations studied by K.O. Friedrichs, first order symmetric hyperbolic systems of the form ut = Aux + Buy ,
(1)
where u(x, y, t) is a vector-valued function, and A and B are real symmetric matrices that may be smooth functions of x and y. The theory of these equations is fairly straightforward: let u(x, y, t) be a solution in the whole (x, y)-space that dies down fast as x and y tend to infinity. Take the scalar product of the equation with u and integrate it over all x and y; we get (u · ut ) dx dy = (u · Aux + u · Buy ) dx dy. (2) If A and B are constant matrices, the integrand on the right is d d (u, Au)/2 + (u, Bu)/2, dx dy
(3)
a sum of perfect x and y derivatives; therefore the integral is zero. The integrand on the left side is the t derivative (1/2)d(u, u)/dt and can be regarded as the t derivative of 1 E(t) = (u, u) dx dy. 2 Since
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