Analysis of a fourth-order compact ADI method for a linear hyperbolic equation with three spatial variables
The authors consider the constant coefficient telegraph equation in a cuboid and build a fourth-order approximation (both in time and in the 3 equidistant spatial step lengths) along with first kind boundary values by a compact factorized difference scheme. They prove the stability of the scheme by a von Neumann analysis where it turns out that the Courant numbers must be restricted by a function of the parameters of the time operator. Under this condition, they also show an error estimate (for sufficiently smooth exact solutions) in a discrete \(H1\)-norm. Since the scheme is factorized, it can be solved by a D'jakonov (-Gunn-Douglas) approach leading to the subsequent solution of a series of tridiagonal systems of linear equations. Numerical experiments show that the approach is indeed more accurate and efficient than the one of several competitors. Let us remark that the imbedding and positive definiteness theorems cited here from a Chinese source from 2011 were known, e.g. in the Soviet literature some 40 years earlier.
alternating direction implicit (ADI) method
scientific article; zbMATH DE number 6168472
scientific article