On the behavior over long time intervals of finite difference and finite element approximations to first order hyperbolic systems

Abstract

Standard error estimates for approximations to hyperbolic equations are only valid over a finite time interval. This paper considers the behavior of the error in the two most common discretization schemes over the entire interval 0 ⩽ t < ∞. Finite difference approximations are analyzed for approximating the Cauchy problem for hyperbolic systems. It is recalled that, if the system is dissipative, error estimates that are global in t can be obtained. For strictly conservative systems, the role of local energy decay is pointed out and used to obtain global in t local in x error estimates. Finite element methods are considered for nonlinear, initial, boundary value problems for hyperbolic systems. It is proven that if the nonlinear term is locally Lipschitz and the equation is dissipative when linearized about the true solution, then global in t convergence follows with the expected rates.