Related papers: Study Notes on Numerical Solutions of the Wave Equ…
Explicit numerical methods based on Lax-Friedrichs and Leap-Frog finite difference approximations are constructed to find the numerical solution of the first-order hyperbolic partial differential equation with point-wise delay or advance,…
Motivated by the problem of solving the Einstein equations, we discuss high order finite difference discretizations of first order in time, second order in space hyperbolic systems.Particular attention is paid to the case when first order…
In this paper, a class of finite difference numerical techniques is presented to solve the second-order linear inhomogeneous damped wave equation. The consistency, stability, and convergences of these numerical schemes are discussed. The…
In this work, we discuss and compare three methods for the numerical approximation of constant- and variable-coefficient diffusion equations in both single and composite domains with possible discontinuity in the solution/flux at…
This paper is part of a program to combine a staggered time and staggered spatial discretization of continuum wave equations so that important properties of the continuum that are proved using vector calculus can be proven in an analogous…
In this paper we give a survey on various multiscale methods for the numerical solution of second order hyperbolic equations in highly heterogeneous media. We concentrate on the wave equation and distinguish between two classes of…
We propose a generalized finite element method for the strongly damped wave equation with highly varying coefficients. The proposed method is based on the localized orthogonal decomposition introduced and is designed to handle independent…
A second order explicit one-step numerical method for the initial value problem of the general ordinary differential equation is proposed. It is obtained by natural modifications of the well-known leapfrog method, which is a second order,…
We develop a hybrid spatial discretization for the wave equation in second order form, based on high-order accurate finite difference methods and discontinuous Galerkin methods. The hybridization combines computational efficiency of finite…
It is known that the digital waveguide (DW) method for solving the wave equation numerically on a grid can be manipulated into the form of the standard finite-difference time-domain (FDTD) method (also known as the ``leapfrog'' recursion).…
Solution of Helmholtz equation with impedance boundary condition on finite interval is equivalently reformulated as steady state of initial boundary value problem for first order hyperbolic system of partial differential equations.…
We use high order finite difference methods to solve the wave equation in the second order form. The spatial discretization is performed by finite difference operators satisfying a summation-by-parts property. The focus of this work is on…
We derive a posteriori error estimates for the the scalar wave equation discretized in space by continuous finite elements and in time by the explicit leapfrog scheme. Our analysis combines the idea of invoking extra time-regularity for the…
In this paper, we present a high order finite difference solver for anisotropic diffusion problems based on the first-order hyperbolic system method. In particular, we demonstrate that the construction of a uniformly accurate fifth-order…
This paper presents a new narrow-stencil finite difference method for approximating the viscosity solution of second order fully nonlinear elliptic partial differential equations including Hamilton-Jacobi-Bellman equations. The proposed…
In this paper, we study a fast and linearized finite difference method to solve the nonlinear time-fractional wave equation with multi fractional orders. We first propose a discretization to the multi-term Caputo derivative based on the…
In this work we explore the fidelity of numerical approximations to the analytic spectra of hyperbolic partial differential equation systems with variable coefficients. We are particularly interested in the ability of discrete methods to…
In this paper we consider the numerical solution of the Hamiltonian wave equation in two spatial dimension. We use the Mimetic Finite Difference (MFD) method to approximate the continuous problem combined with a symplectic integration in…
This paper presents a new finite difference algorithm for solving the 2D one-way wave equation with a preliminary approximation of a pseudo-differential operator by a system of partial differential equations. As opposed to the existing…
We extend our previous work [F. Henr'iquez and J. S. Hesthaven, arXiv:2403.02847 (2024)] to the linear, second-order wave equation in bounded domains. This technique uses two widely known mathematical tools to construct a fast and efficient…