Finite difference method

Abstract: In this paper, new explicit methods for the finite difference solution of a parabolic partial differential equation are derived. The new methods use stable asymmetric approximations to the partial differential equation which when coupled in groups of 2 adjacent points on the grid result in implicit equations which can be easily converted to explicit form which in turn offer many advantages. By judicious use of alternating this strategy on the grid points of the domain results in an algorithm which possesses unconditional stability. The merit of this approach results in more accurate solutions because of truncation error cancellations. The stability, consistency, convergence and truncation error of the new method is discussed and the results of numerical experiments presented.

Abstract: 1 Introduction: 1.1 Numerical solution process. 2 Computer aided design in magnetics: 2.1 Finite element based CAD systems 2.2 Design strategies. 3 Electromagnetic fields: 3.1 Quasi stationary fields 3.2 Boundary value problem 3.3 Field equations in partial differential form. 4 Potentials and formulations: 4.1 Magnetic vector potential 4.2 Electric vector potential for conducting current 4.3 Electro-static scalar potential 4.4 Magnetic scalar potential 4.5 A? -formulation 4.6 AV-formulation 4.7 In-plane formulation 4.8 AV-formulation with v?B motion term 4.9 Gauge conditions 4.10 Subsequent treatment of the Maxwell equations. 5 Field computation and numerical techniques: 5.1 Magnetic equivalent circuit 5.2 Point mirroring method 5.3 The numerical solution of partial differential equations 5.4 Finite difference method 5.5 Finite element method 5.6 Material modelling 5.7 Numerical implementation of the FEM 5.8 Adaptive refinement for 2D triangular meshes 5.9 Coupling of field and circuit equations 5.10 Post-processing. 6 Coupled field problems: 6.1 Coupled fields 6.2 Strong and weak coupling 6.3 Coupled problems 6.4 Classification of coupled field problems. 7 Numerical optimisation: 7.1 Electromagnetic optimisation problems 7.2 Optimisation problem definition 7.3 Methods. 8 Linear system equation solvers: 8.1 Methods 8.2 Computational costs. 9 Modelling of electrostatic and magnetic devices: 9.1 Modelling with respect to the time 9.2 Geometry modelling 9.3 Boundary conditions 9.4 Transformations. 10 Examples of computed models: 10.1 Electromagnetic and electrostatic devices 10.2 Coupled thermo-electromagnetic problems 10.3 Numerical optimisation