Nonlinear Monotonization of the Babenko Scheme

Abstract

The goal of the paper is to present and test the nonlinear monotonization of the Babenko scheme for solving 2D linear advection equation with alternating‐sign velocities. The numerical method of monotonization is based on the idea of limited artificial diffusion. There are some approaches for constructing quasi‐monotonic second order approximation schemes for solving hyperbolic systems and equations of gas dynamics: flux correction methods, the Godunov method, TVD methods and others. In particular, many authors developed the idea of TVD method. We try to use this idea to get a new quasi‐monotonic high order accuracy scheme based on the well‐known non‐monotonic Babenko scheme. The algorithm is presented for 1D problem. For testing 2D problem we use the splitting algorithm. The proposed monotonized scheme has shown the best results among all considered in the paper schemes especially for non-smooth initial profile.

Nonlinear Monotonization of the Babenko Scheme for the Quasi‐linear Advection Equation

“…In the previous papers [4,5,6,7,8,9] authors have investigated theoretically and tested experimentally 26 different finite-difference schemes on 4 point patterns for the simplest hyperbolic equation: linear advection equation. This equation has the main features of every hyperbolic equation and is the important part of many mathematical models.…”

“…In [4,5,6,7,8,9] some new schemes were constructed for solving this advection equation. The nonlinear monotone Babenko scheme ("square") proved to be the best among all 26 schemes.…”