Numerical Solution to the Cargo LeRoux Model

This article presents the Godunov method applied to solve the Riemann problem for the isentropic Cargo LeRoux model. The pressure-density relation for this model yields Riemann invariants that are complicated and computationally challenging to process. Only the exact solution has been developed for this model, and no numerical method has been applied thus far. Incorporating many computational checks is necessary for the numerical simulations of this model. Hence, it is ensured that variables and related quantities in the pressure equation remain positive, and the integrals converge within given accuracy conditions without casting complex values. Two nonlinear equations are obtained for possible wave configurations in the exact solutions. This system of two nonlinear algebraic equations is solved with the Newton-Raphson method. This root-finding process is sensitive to the initial guess values. A special algorithm is used to guess the initial values for the Newton-Raphson procedure. It is based on evaluating the residual functions and finding appropriate initial guesses by sorting the answers. These steps effectively simulate complex terms. The system is hyperbolic, with three distinct eigenvalues of the Jacobian matrix, and the solution consists of three waves. The tests showed that the Godunov scheme is applicable and achieves good accuracy for this model. However, the central scheme fails to solve this problem. This study bridges the gap between model analysis and developing numerical methods for solving isentropic Euler equation models.