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