An accurate, stable and efficient domain-type meshless method for the solution of MHD flow problems

• Published in: Journal of computational physics Vol. 228; no. 21; pp. 8135 - 8160
• Main Authors: Bourantas, G.C., Skouras, E.D., Loukopoulos, V.C., Nikiforidis, G.C.
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Inc 20.11.2009; Elsevier
• Subjects: ALGORITHMS; APPROXIMATIONS; BOUNDARY CONDITIONS; Computational techniques; CONVERGENCE; CROSS SECTIONS; DIRICHLET PROBLEM; Exact sciences and technology; EXACT SOLUTIONS; FUNCTIONS; GEOMETRY; HARTMANN NUMBER; LEAST SQUARE FIT; MAGNETOHYDRODYNAMICS; MATHEMATICAL METHODS AND COMPUTING; Mathematical methods in physics; Meshless; MHD; MLS; Physics; Point collocation; MLS; MHD; 47.11.−j; 02.60.−x; 52.30Cv; 47.65.−d; Point collocation; Hartmann number; Meshless; Operator; Magnetohydrodynamics; Boundary conditions; Robin problem; 02.60.-x; Calculation methods; Exact solution; MHD flow; Algorithms; Numerical solution; Meshless method; Dirichlet problem; Calculation; 47.11.-j; Cross sections; 47.65.-d
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2009.07.031

The aim of the present paper is the development of an efficient numerical algorithm for the solution of magnetohydrodynamics flow problems for regular and irregular geometries subject to Dirichlet, Neumann and Robin boundary conditions. Toward this, the meshless point collocation method (MPCM) is used for MHD flow problems in channels with fully insulating or partially insulating and partially conducting walls, having rectangular, circular, elliptical or even arbitrary cross sections. MPC is a truly meshless and computationally efficient method. The maximum principle for the discrete harmonic operator in the meshfree point collocation method has been proven very recently, and the convergence proof for the numerical solution of the Poisson problem with Dirichlet boundary conditions have been attained also. Additionally, in the present work convergence is attained for Neumann and Robin boundary conditions, accordingly. The shape functions are constructed using the Moving Least Squares (MLS) approximation. The refinement procedure with meshless methods is obtained with an easily handled and fully automated manner. We present results for Hartmann number up to 10 5 . The numerical evidences of the proposed meshless method demonstrate the accuracy of the solutions after comparing with the exact solution and the conventional FEM and BEM, for the Dirichlet, Neumann and Robin boundary conditions of interior problems with simple or complex boundaries.