Positivity-preserving pseudo arc-length method for solving extreme explosive shock wave flow field problems in hyperbolic conservation law equations

Hyperbolic conservation law equation is a system of partial differential equations, which has the characteristics of hyperbolic type, and its solution will eventually evolve into a singular solution with strong discontinuity and weak discontinuity. In solving hyperbolic conservation law equations, in addition to eliminating or reducing the singularity of the equation, it is also involved in solving the problems of high density ratios, high pressure ratios, low densities, and low pressures in extreme explosive shock wave flow field. The numerical solution process often produces negative density, negative pressure, or negative internal energy, which in turn triggers nonlinear instabilities in the region of large gradients, eventually leading to termination of the computational procedure. To effectively solve the above difficulties and improve the robustness of the algorithm, this paper proposes a high-order accuracy pseudo arc length method (PALM) with positivity-preserving for solving hyperbolic conservation law equations. The core idea of the algorithm introduces arc length constraint equations to realize the smooth transition of physical quantities with large gradients in the original physical space in the computational arc length space, so as to eliminate or attenuate the singularity of the discontinuous solutions in the computational process. In the orthogonal pseudo arc length coordinate system, a new weighted essentially non-oscillatory-z scheme with a global smoothness indicator is used for high-order reconstruction, effectively addressing the issue of nonphysical oscillations caused by high-order schemes. In addition, the positivity-preserving limiter of the high-order pseudo arc length adaptive method in the arc length computational coordinate system is further constructed and proved, and the grid motion limiting strategy is designed, which greatly enhances the reliability and robustness of the algorithm. Finally, numerical test results show that the PALM effectively reduces numerical oscillations and achieves high-resolution capture of discontinuity problems.