Error analysis and numerical simulation of magnetohydrodynamics (MHD) equation based on the interpolating element free Galerkin (IEFG) method. (English) Zbl 1412.65070
Summary: The MHD equation has some applications in physics and engineering. The main aim of the current paper is to propose a new numerical algorithm for solving the MHD equation. At first, the temporal direction has been discretized by the Crank-Nicolson scheme. Also, the unconditional stability and convergence of the time-discrete scheme have been investigated by using the energy method. Then, an improvement of element free Galerkin (EFG), i.e., the interpolating element free Galerkin method has been employed to discrete the spatial direction. Furthermore, an error estimate is presented for the full discrete scheme based on the Crank-Nicolson scheme by using the energy method. We prove that convergence order of the numerical scheme based on the new numerical scheme is \(\mathcal{O}(\tau^2 + \delta^m)\). In the considered method the appeared integrals are approximated using Gauss Legendre quadrature formula. Numerical examples confirm the efficiency and accuracy of the proposed scheme.
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 76W05 | Magnetohydrodynamics and electrohydrodynamics |
| 35Q35 | PDEs in connection with fluid mechanics |
| 65M15 | Error bounds 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 |
| 65D32 | Numerical quadrature and cubature formulas |
Keywords:
magnetohydrodynamics (MHD) equation; element free Galerkin (EFG) method; interpolating moving least squares approximation; unconditional stability; convergence; interpolating element free Galerkin techniqueReferences:
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