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On stable implicit difference scheme for hyperbolic-parabolic equations in a Hilbert space. (English) Zbl 1175.65103
For a self-adjoint, positive-definite operator $$A$$ the authors consider the differential equation $$u' + Au = g$$ for $$-1 < t < 0$$ followed by $$u'' + Au = f$$ for $$0 < t < 1$$. The initial condition at $$t = -1$$ depends linearly on future values of $$u$$ for $$0 < t < 1$$. The authors analyze the stability of an implicit discretization.

##### MSC:
 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 35M13 Initial-boundary value problems for PDEs of mixed type 34G10 Linear differential equations in abstract spaces 65L12 Finite difference and finite volume methods for ordinary differential equations 65L20 Stability and convergence of numerical methods for ordinary differential equations 65J08 Numerical solutions to abstract evolution equations
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