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Numerical analysis and simulation for a nonlinear wave equation. (English) Zbl 1331.65142
Summary: In this work we study a nonlinear wave equation, depending on different norms of the initial conditions, which has a bounded solution for all \(t > 0\) or \(0 < t < T_0\) for some \(T_0 > 0\). We also prove that the solution may blow-up at \(T_0\). Proofs of some the analytical results listed are sketched or given. For approximate numerical solutions we use the finite element method in the spatial variable and the finite difference method in time. The nonlinear system for each time step is solved by Newton’s modified method. We present a numerical analysis for error estimates and numerical simulations to illustrate the convergence of the theoretical results. We present too, the singularity points \((x^\ast, t^\ast)\), where the blow-up occurs for different \(\rho\) values in a numerical simulation.

MSC:
65M60 Finite element, Rayleigh-Ritz and Galerkin 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
35L70 Second-order nonlinear hyperbolic equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
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