An Efficient Characteristic Method for the Magnetic Induction Equation with Various Resistivity Scales

Abstract

Abstract. In this paper, we develop an efficient characteristic finite element method (FEM) for solving the magnetic induction equation in magnetohydrodynamics (MHD). We carry out numerical experiments on a two dimensional test case to investigate the influence of resistivity at different scales. In particular, our numerical results exhibit how the topological structure and energy of the magnetic field evolve for different scales of resistivity. Magnetic reconnection can also be observed in the numerical experiments.

Simulation of Multiphysics Multiscale Systems: Introduction to the ICCS’2007 Workshop

“…The second essential component for numerical simulation of multiphysics and multiscale problems includes efficient numerical algorithms and advanced computational techniques. Computational methodologies and programming tools [19][20][21] and advanced mathematical and numerical algorithms [22][23][24][25] are indispensable for efficient implementation of multiscale multiphysics models, which are computationally very intensive and often intractable using ordinary methods. Cellular automata [11,26,27] and the lattice Boltzmann method [28][29][30][31], which can be considered both as modeling tools and as numerical techniques, prove to be very powerful and promising in modeling complex flows and other multiscale complex systems.…”

“…Advanced mathematical and numerical algorithms are required for effective coupling of various models across multiple scales and for efficient reduction of the computations needed for fine scale simulations without loss of accuracy [22][23][24][25]. As a significant extension to the classical multiscale finite element methods, paper [24] is devoted to the development of a theoretical framework for multiscale Discontinuous Galerkin (DG) methods and their application to efficient solutions of flow and transport problems in porous media with interesting numerical examples.…”

“…In this work, local DG basis functions at the coarse scale are first constructed to capture the local properties of the differential operator at the fine scale, and then the DG formulations using the newly constructed local basis functions instead of conventional polynomial functions are solved on the coarse scale elements. In [23], an efficient characteristic finite element method is proposed for solving the magnetic induction equation in magnetohydrodynamics, with numerical results exhibiting how the topological structure and energy of the magnetic field evolve for different resistivity scales. Paper [22] includes a fast Fourier spectral technique to simulate the Navier-Stokes equations with no-slip boundary conditions, enforced by an immersed boundary technique called volume-penalization.…”