Convergence of numerical algorithms for semilinear hyperbolic systems
The paper is concerned with blowing up solutions of semilinear hyperbolic systems and their numerical approximation. It is recalled that such systems, of the form \[ \partial_t u + A \partial_x = f(u) \] with \(A\) diagonal(izable) on \(R\) and \(f\) quadratic, admit a unique bounded solution for initial data in \(W^{1,\infty}\), until a (possibly bounded) blow up time \(T^*\). The authors first show an ``abstract'' result. They consider a system of the previous form and its discretization by an explicit finite difference scheme. Under suitable assumptions (called local uniform stability and local uniform consistency), they show that the scheme converges uniformly on \([0,T_0]\) for \(T_0<T^*\) and that the maximal convergence time coincides with \(T^*\). Then they study a semi-implicit scheme and show a similar result. Subsequently, they focus on the approximation of semilinear systems of wave equations. They construct a scheme that satisfies the stability and consistency assumptions. Finally, numerical experiments are presented. They concern the Broadwell system, \(2\times 2\) systems (either conservative or not), and systems of wave equations (with an insight into \(3\)-dimensional problems). They are in good accordance with the theory.
scientific article; zbMATH DE number 1395899
scientific article