Method of Lines Transpose: An Efficient Unconditionally Stable Solver for Wave Propagation

Abstract

Building upon recent results obtained in [7,8,9], we describe an efficient second order, A-stable scheme for solving the wave equation, based on the method of lines transpose (MOL T ), and the resulting semi-discrete (i.e. continuous in space) boundary value problem. In [7], A-stable schemes of high order were derived, and in [9] a high order, fast O(N ) spatial solver was derived, which is matrix-free and is based on dimensional-splitting.In this work, are interested in building a wave solver, and our main concern is the development of boundary conditions. We demonstrate all desired boundary conditions for a wave solver, including outflow boundary conditions, in 1D and 2D. The scheme works in a logically Cartesian fashion, and the boundary points are embedded into the regular mesh, without incurring stability restrictions, so that boundary conditions are imposed without any reduction in the order of accuracy. We demonstrate how the embedded boundary approach works in the cases of Dirichlet and Neumann boundary conditions. Further, we develop outflow and periodic boundary conditions for the MOL T formulation. Our solver is designed to couple with particle codes, and so special attention is also paid to the implementation of point sources, and soft sources which can be used to launch waves into waveguides.

Method of Lines Transpose: Energy Gradient Flows Using Direct Operator Inversion for Phase-Field Models

“…It remains to discretize the convolution integral (18), and obtain a fully discrete algorithm. In our previous works [4,7,5,6], we accomplish this with fast convolution; for convenience, we will summarize the fast algorithm here. First, the particular solution (19) is split into…”

“…We now extend the 1D solver to multiple spatial dimensions via dimensional splitting [7,5,6]. We first write the 2D modified Helmholtz operator as…”

A particle-in-cell method for the simulation of plasmas based on an unconditionally stable field solver

“…Under the Lorenz gauge condition, Maxwell's equations reduce to uncoupled wave equations for the scalar and vector potentials, Φ and A. Recently, a novel method for the solution of the wave equation has been developed [4,5,6], based on the Method of Lines Transpose (MOLT), dimensional splitting and an efficient 1D integral solution method, which is unconditionally stable (or A-stable) -that is, it is not subject to the Courant-Friedrichs-Lewy (CFL) restriction limiting the ratio of the temporal step size to the spatial step size, typical of widely used explicit methods. In this work, we apply this method to the uncoupled wave equations for Φ and A to solve Maxwell's equations with a method comparable in computational cost and complexity of code to explicit methods such as the well-known finite difference time-domain (FDTD) method (also known as the Yee scheme) [7,8], but without introducing a CFL restriction based on the speed of light as in such explicit methods.…”

“…In this work, we obtain solutions that satisfy exactly discrete forms of Gauss' law for the electric field and the divergence-free condition for the magnetic field through a staggered grid approach, adapted from the well-known Yee grid [7], with a Poisson equation formulation for the scalar potential. In addition to eliminating the CFL restriction, the wave solver method used in this work offers the handling of complex boundary geometry in a Cartesian grid, demonstrated for the scalar wave equation to second-order accuracy in [6], without using the staircasing approximation of traditional FDTD methods [10], and can be extended to higher-order accuracy [5], features that will be incorporated into our PIC method in future work.…”

“…However, we can still achieve improved wave propagation properties with our method by using higher-order accurate versions of the wave solver, as demonstrated in [5], allowing use of CFL numbers of 20-100+ (naturally with some loss of time accuracy as the time step size increases). Appropriate filtering techniques, perhaps using the artificial dissipation introduced in [6] or an adaptive variation thereof or perhaps using other standard techniques [31], could be applied to suppress numerical Cerenkov instabilities in simulations where they arise. Further, our potential-based formulation results in the uncoupling of the four components of the scalar and vector potential and with each wave equation having only one component of the charge density or current density appearing as a source, whereas the PSATD curl equations couple the six components of the electric and magnetic fields, with many components of the current density appearing in each equation (due to k × J terms).…”