The TVD modification of third-order Godunov method

A study of the monotonicity properties of a new modification of the Godunov method, which has a third order of approximation in space and time, is presented. The difference scheme of the method uses a simultaneous discretization of conservation laws in space and time without of Runge - Kutta stages. This method is a development of the second order Godunov method through the connection of two additional fluxes correction procedures. The first procedure increases the order of approximation for linear systems of equations. The second procedure uses the finite differences of the Jacobi matrix of the system of equations and eliminates the second-order error that occurs due to the nonlinearity of the equations. The TVD property of the difference scheme is strictly proved in relation to the linear scalar transfer equation when using an generalized adaptive limiter of central differences. New modifications of limiters are proposed that significantly improve the accuracy of the solution in the vicinity of discontinuities and local extremes. New limiters are obtained from the known ones using a simple function of local smooth deformation. An economical version of the third-order Godunov method for gas dynamics equations with quadratic reconstruction of parameters in primary variables is presented. The use of primary variables for reconstruction significantly reduces the FPU time during calculations. The advantages of the third-order method in terms of the accuracy of the solution in the vicinity of contact discontinuities and the boundaries of the rarefaction wave are demonstrated on standard tests. The proposed approach to constructing third-order difference schemes can be used for inhomogeneous and two-dimensional hyperbolic systems of nonlinear equations.