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Efficient difference method for unsteady problem with small parameter at high derivative. (Russian) Zbl 0755.65092

A several component algorithm for the numerical solution of the parabolic-hyperbolic problem \({\partial u\over\partial t}=\alpha{it \partial^ 2u\over\partial x^ 2}-a{\partial u\over\partial x}+f(x,t)\), \((x,t)\in\Omega\), \(\Omega:=\{-\infty<x<\infty\), \(t\geq 0\}\), when \(a\gg\alpha\) is proposed and analyzed. The main idea is as follows: two meshes are defined; \(\omega_{h\tau}^{(1)}:=\{(x_ i,t_ j)\), \(x_ i=ih\), \(i=\dots,-1,0,1,\dots,t_ j=j\tau\), \(j=0,1,\dots\}\) and \(\omega^{(2)}_{h\tau}=\{x_{i+1/2},t_ j\}\).
The discretization of the original problem is given on this double mesh using the functions \(y_ 1=y_ 1(x,t)\), \((x,t)\in\omega^{(1)}_{h\tau}\), \(y_ 2=y_ 2(\tilde x,t)\), \((\tilde x,t)\in\omega^{(2)}_{h\tau}\) as components. The difference scheme is: \(y_{1t}=\alpha\hat y_{1\bar xx}-ay_{2\bar x}+f\), \(\bar y_{2t}=\alpha\hat y_{1\bar xx}-a\hat y_{2\bar x}+\hat f\), where \(\bar y_ 2=1/2[y(x_{i-1/2})+y(x_{i+1/2})]\). A four component difference scheme is given, too.
The stability of this finite difference scheme is proved and illustrated by numerical experiments.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35M10 PDEs of mixed type
35K15 Initial value problems for second-order parabolic equations
35L15 Initial value problems for second-order hyperbolic equations
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