Various Numerical Algorithms for Solving Partial Differential Equation Systems Arising from Shock Tube Problems and Similar Applications

The shock tube problem serves as a classical benchmark in computational fluid dynamics, designed to test numerical methods' capability in handling shocks and contact discontinuities. This problem type is typically governed by the Euler equations, which constitute a system of nonlinear hyperbolic partial differential equations. Solving these problems requires specialized numerical algorithms capable of handling potential shock waves and discontinuous solutions. Common numerical algorithms can be classified into the following categories: Finite Difference Methods represent traditional approaches that require sufficiently smooth solution domains, but tend to produce oscillations when dealing with discontinuities like shocks. To address this, artificial viscosity methods were developed, which suppress oscillations by adding artificial dissipation terms, though this may compromise solution accuracy. In code implementation, this typically involves adding a diffusion term proportional to the second derivative of the solution variables. Finite Volume Methods have become the mainstream approach, directly discretizing the integral form of conservation laws to naturally maintain solution conservativity. The Godunov method serves as a typical representative, utilizing exact or approximate Riemann solutions to compute fluxes at cell interfaces, demonstrating excellent shock-capturing capabilities. Implementation-wise, this involves reconstructing primitive variables at cell interfaces and solving local Riemann problems. High-Resolution Schemes employ limiter techniques to balance accuracy and stability, including MUSCL, TVD, ENO, and WENO formats. These schemes automatically reduce order near shocks to prevent oscillations while maintaining high-order accuracy in smooth regions. The WENO scheme is particularly suitable for multi-scale problems, implementing weighted combinations of different stencil approximations through nonlinear weights that adapt to solution smoothness. Additionally, Discontinuous Galerkin-based finite element methods offer both high precision and geometric flexibility, though with higher computational costs. These methods use piecewise polynomial approximations within each element and employ numerical fluxes at element boundaries. Recently developed compact schemes and hybrid methods have also demonstrated advantages in specific problems, often combining different discretization techniques for optimal performance. When selecting algorithms, considerations must include computational efficiency, accuracy requirements, and problem characteristics. For problems involving strong shocks, conservative high-resolution finite volume methods are typically the preferred choice, often implemented with flux limiters and appropriate Riemann solvers in practical CFD codes.