Development of new and modified numerical methods for hyperbolic conservation laws

Abstract

The aim of the present work is to propose and modify several numerical methods including the classes of temporal discretization methods for hyperbolic conservation laws. The first order in space standard Lax approximation is updated to modified first-order and newly proposed third-order accurate approximation. Presently proposed methods can be coupled with the modified and newly proposed Lax approximations and this coupling make the methods conservative. Some additional new classes of explicit and implicit methods for PDEs in time are proposed. Additionally, some new methods are given to reduce oscillations in the solutions. These new methods of reducing oscillations provide the conditions for coupling of first and higher-order methods.