On an essentially conservative scheme for hyperbolic conservation laws (Q1331438)

This paper is concerned with the numerical solution of scalar equations and systems of conservation laws. In the first part of the paper the author proposes an essentially conservative second-order scheme, proving that it satisfies the so called total variation bounded (TVB) property. Such a property is a weaker requirement than the total variation diminishing property considered e.g. by \textit{J. B. Goodman} and \textit{R. J. LeVeque} [Math. Comput. 45, 15-21 (1985; Zbl 0592.65058)] which restricts severely the order of convergence of the schemes. For the scalar conservation law, a detailed theoretical study is carried out in Section 2. Next, the author extends the result of the scalar case to systems of conservation laws by using some kind of averaging, such as \textit{P. L. Roe's} averaging [J. Comput. Phys. 43, 357-372 (1981; Zbl 0474.65066)]. Again a second-order essentially conservative scheme is proposed. In the last section the numerical results obtained by solving numerically several problems with the proposed schemes are presented. As scalar problems a linear equation with periodic initial conditions and a shock wave problem for the inviscid Burgers' equation are considered. As systems of equations, Euler's equations of gas dynamics with discontinuous initial values are considered (here an artificial compression technique is introduced in order to improve the resolution of shock waves). It is shown that the numerical solutions provided by the methods follow closely the behaviour of the exact solutions.