Finite difference method

Abstract: A new formulation is presented for the systematic development of perfectly matched layers (PML) from Maxwell's equations in properly constructed anisotropic media. The proposed formulation has an important advantage over the original Berenger's PML in that it can be implemented in the time domain without any splitting of the fields. Results from 3D simulations illustrate the effectiveness of the proposed method.

Abstract: The system of nonlinear partial differential equations governing the transient motion of a cable immersed in a fluid is solved by finite difference methods. This problem may be considered a generalization of the classical vibrating string problem in the following respects: a) the motion is two dimensional, b) large displacements are permitted, c) forces due to the weight of the cable, buoyancy, drag and virtual inertia of the medium are included, and d) the properties of the cable need not be uniform. The numerical solution of this system of equations presents a number of interesting mathematical problems related to: a) the nonlinear nature of the equations, b) the determination of a stable numerical procedure, and c) the determination of an effective computational method. The solution of this problem is of practical significance in the calculation of the transient forces acting on mooring and towing lines which are subjected to arbitrarily prescribed motions. 1. Introduction. This problem arose as a result of an urgent requirement by the Navy in connection with a series of nuclear explosion tests which were conducted in the Pacific. In preparation for these tests a number of ships were instrumented and moored at specified locations from the explosion point. These positions had to be maintained intact during the period preceding the explosion. However, the bobbing up and down of the ships due to ocean waves could excite transient forces in the mooring lines sufficient to break them and thus result in the loss of informa- tion from the tests. Several months prior to these tests a request was made to the Applied Mathematics Laboratory to calculate the magnitude of the forces acting on the mooring lines for waves of varying amplitude and frequency. The two factors which made a theoretical solution feasible at this time, whereas it would not have been possible several years ago, 'vere: a) the availability of a high-speed computer and b) the recent progress made in the understanding and development of nu- merical methods for the solution of systems of partial differential equations by finite-difference methods. Although this problem was solved to satisfy a specific request, it is more useful to regard it as the general problem of the two-dimensional motion of a cable or rope immersed in a fluid, and it becomes immediately apparent that its solution i.s applicable to a wide class of engineering problems involving the motion of cables. such as: a) the laying of submarine telegraph cables, b) the towing of a ship or other object in water, or c) the snapping of power lines as a result of transient forces caused by storms. The problem may be stated abstractly as follows: Given the initial conditions (i.e., position and velocity at any time, t0) and boundary

Abstract: New multigrid methods are developed for the maximum principle preserving immersed interface method applied to second order linear elliptic and parabolic PDEs that involve interfaces and discontinuities. For elliptic interface problems, the multigrid solver developed in this paper works while some other multigrid solvers do not. For linear parabolic equations, we have developed the second order maximum principle preserving finite difference scheme in this paper. We use the Crank--Nicolson scheme to deal with the diffusion part and an explicit scheme for the first order derivatives. Numerical examples are also presented.