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On the second order of accuracy difference scheme for hyperbolic equations in a Hilbert space. (English) Zbl 1098.65055
Following abstract initial value problem is considered \[ {{d^2u(t)}\over{dt^2}}+A(t)u(t)=f(t),\;\;0\leq t\leq T,\;\;u(0)=\psi,\;\;u^{'}(0)=\psi,(1) \] where \(A(t)\) is a self-adjoint, positive definite operator in a Hilbert space \(H\), with a \(t\)-independent domain \(D=D[A(t)]\). The authors establish for the problem (1) a second-order finite difference two-level scheme with the time step \(\tau\) and the discrete solution \(u_k\). Two theorems on stability in the discrete maximum norm are proven:
Under the assumption that \(u(0)\in D[A^{1\over2}(0)]\) the unconditional stability concerning \[ u_k\;\;\text{ and}\;\;{{u_k-u_{k-1}}\over\tau}; \]
under the assumption that \(u(0)\in D[A(0)]\) and \(u^{'}(0)\in D[A^{1\over2}(0)]\) the unconditional stability concerning \[ u_k,\;\;{{u_k-u_{k-1}}\over\tau}\;\;\text{ and}\;\;{{u_{k-1}-2u_k+u_{k+1}}\over{\tau^2}}. \]

MSC:
65J10 Numerical solutions to equations with linear operators
65L20 Stability and convergence of numerical methods for ordinary differential equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
34G10 Linear differential equations in abstract spaces
35L90 Abstract hyperbolic equations
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