Stability analysis of spectral methods for hyperbolic initial-boundary value systems. (English) Zbl 0628.65109

Gottlieb, David; Lustman, Liviu; Tadmor, Eitan

SIAM J. Numer. Anal. 24, 241-256 (1987).

The authors consider the system of first order hyperbolic partial differential equations with constant coefficients: \(u_ t=Au_ x,\) \(- 1<x<1\), \(t>0\), where \(u=u(x,t)=(u^ i(x,t),\quad u^{ii}(x,t))\) is an n-vector and \(A=diag(A^ i,A^{ii}),\quad A^ i=diag(a_ 1,...,a_{\ell},)\) \(a_ i<0\) \((i=1,...,\ell)\), \(A^{ii}=diag(a_{\ell +1},...,a_ n),\) \(a_ j>0\) \((j=\ell +1,...,n)\), with zero initial conditions and boundary conditions: \(u^ i(-1,t)=Lu^{ii}(-1,t)+g^ i(t),\) \(u^{ii}(1,t)=Ru^ i(1,t)+g^{ii}(t).\) Here, \(g^ i\) and \(g^{ii}\) are given \(\ell\)-vector and (n-\(\ell)\)-vector, respectively; \(u^ i\) and \(u^{ii}\) are \(\ell\)- and (n-\(\ell)\)-components of u, respectively; L and R are constant matrices of \(\ell \times (n-\ell)\) and (n-\(\ell)\times \ell\) types, respectively. They present sufficient conditions for stability of spectral methods of approximation for the above system.

Reviewer: R.Iino

MSC:

65N35	Spectral, collocation and related methods for boundary value problems involving PDEs
35L50	Initial-boundary value problems for first-order hyperbolic systems

Keywords:

dissipative boundary conditions; Chebyshev and Legendre collocations; first order system; stability; spectral methods