Localized meshless point collocation method for time-dependent magnetohydrodynamics flow through pipes under a variety of wall conductivity conditions

In this article a numerical solution of the time dependent, coupled system equations of magnetohydrody- namics (MHD) flow is obtained, using the strong-form local meshless point collocation (LMPC) method. The approxima- tion of the field variables is obtained with the moving least squares (MLS) approximation. Regular and irregular nodal distributions are used. Thus, a numerical solver is developed for the unsteady coupled MHD problems, using the collo- cation formulation, for regular and irregular cross sections, as are the rectangular, triangular and circular. Arbitrary wall conductivity conditions are applied when a uniform mag- netic field is imposed at characteristic directions relative to the flow one. Velocity and induced magnetic field across the section have been evaluated at various time intervals for sev- eral Hartmann numbers (up to 105) and wall conductivities. The numerical results of the strong-form MPC method are compared with those obtained using two weak-form mesh- less methods, that is, the local boundary integral equation (LBIE) meshless method and the meshless local Petrov– Galerkin (MLPG) method, and with the analytical solutions, where they are available. Furthermore, the accuracy of the method is assessed in terms of the error norms L 2 and L ∞ , the number of nodes in the domain of influence and the time step length depicting the convergence rate of the method. Run time results are also presented demonstrating the efficiency and the applicability of the method for real world problems.