A new alternating‐direction finite element method for hyperbolic equation

Abstract

A modified backward difference time discretization is presented for Galerkin approximations for nonlinear hyperbolic equation in two space variables. This procedure uses a local approximation of the coefficients based on patches of finite elements with these procedures, a multidimensional problem can be solved as a series of one-dimensional problems. Optimal order H 1 0 and L 2 error estimates are derived.

High Efficient Numerical Methods for Viscous and Nonviscous Wave Problems

“…Since there are many efficient algorithms known for them, we exclude their consideration. Some conventional numerical approaches for solving wave equations introduce auxiliary variables to rewrite the equations as first-order hyperbolic systems [1][2][3]; however, these approaches introduce new unknowns which results in an increase in the number of variables in the discrete problems. Thus, there are advantages in keeping the formulation (1)-(3) involving the second time-derivative and a scalar unknown.…”

“…The wave equation is often solved by explicit time-stepping schemes, which require to choose a time step size sufficiently small to satisfy the stability condition and to reduce numerical dispersion as well. As is well known, the alternating direction implicit (ADI) schemes [3][4][5][6][7][8][9][10][11][12][13][14] are unconditionally stable and only need to solve a sequence of tridiagonal linear systems. In 1955, Peaceman and Rachford Jr. [15] presented a method to solve two-dimensional parabolic equations.…”

“…Thus, there are good reasons to try to keep the formulation involving the second time-derivative and a scalar unknown. Lim et al [11] introduce a stable three-level locally one-dimensional (LOD) method for solving (1)- (3), which is second-order in space.…”

“…where = 0, , = 1, − 1. In addition, following the idea of Douglas [2,3], we can get a compact Douglas scheme as follows:…”

A new fourth‐order numerical algorithm for a class of three‐dimensional nonlinear evolution equations

“…It is well known that ADI methods can change a multidimensional problem into a series of independent 1‐D problems and have an advantage in saving computational cost (cf. [7, 13, 24–30]). Recently, various HOC ADI methods [7, 24–29], which preserve the computational efficiency of ADI approaches and high‐order accuracy of HOC schemes, have been developed for linear and nonlinear time‐dependent problems.…”