A note on the second order of accuracy difference schemes for hyperbolic--parabolic equations
The nonlocal boundary value problem for hyperbolic-parabolic equations \[ \begin{cases} \frac{{d^2 u(t)}}{{dt^2}} + Au(t) = f(t), & {0 \leq t \leq 1}, \\ \frac{{du(t)}}{{dt}} + Au(t) = g(t), & {- 1 \leq t \leq 0}, \\ u({-1}) = \alpha u(\mu) + \varphi , & 0 \leq \alpha \leq 1,\;\;0 \leq \mu \leq 1. \end{cases} \] in a Hilbert space \(H\) with self-adjoint positive definite operator \(A\) is considered. Second order of accuracy difference schemes approximately solving this boundary value problem are presented. Stability estimates for the solution of these difference schemes are established. In applications, stability estimates for the solutions of the difference schemes of mixed type boundary value problems for hyperbolic-parabolic equations are obtained. Theoretical statements for the solution of these difference schemes for hyperbolic-parabolic equation are supported by the results of numerical experiments.
second order of accuracy difference schemes
scientific article; zbMATH DE number 2175470