An Analysis of a Family of Difference Schemes for Solving Hyperbolic Partial Differential Equations

Partial differential equations are an integral part of modern scientific development. Hyperbolic partial differential equations are encountered in many fields and have many applications—both linear and nonlinear types, with some being semilinear and quasilinear. In this paper, a family of implicit numerical schemes for solving hyperbolic partial differential equations is derived, utilizing finite differences and tridiagonal sweep. Through the discrete Fourier transform, a necessary and sufficient condition for convergence is proven for the linear version of the family of difference schemes, expanding the known results on boundary conditions that ensure convergence. Numerical verification confirms the found condition. A series of experiments on different boundary conditions and semilinear hyperbolic PDEs show that the same condition seems to also hold in those cases. In view of the results, an optimal subset of the family is found. A comparison between the implicit schemes and an explicit analogue is conducted, demonstrating the gained efficiency of the implicit schemes.