A balanced algorithm of the hybrid large-particle method and its verification on some test problems

In series of numerical experiments involving the Shu-Osher problem of nonlinear acoustics and the Woodward-Colella problem of two interacted blast waves, the author studied the computational properties of a new algorithm for the hybrid large-particle method. The numerical method is a two-step predictor-corrector type in time. Spatial derivatives are split by physical processes. At the first stage of splitting, the gradient and deformation terms of the conservation laws are taken into account, and at the second stage, convective flows are taken into account. The proposed balanced algorithm of the method includes a more dissipative upwind reconstruction of fluxes at the “predictor” step and a centered (nondissipative on smooth solutions) approximation at the correction step: CDP2-UC (Customizable Dissipative Properties — Upwind-Centered). For a more flexible control of the numerical viscosity, a nonlinear correction of the scheme based on a parametric combination of known limiters is implemented. The numerical scheme has a second-order approximation in space and time on smooth solutions. The balanced algorithm of the hybrid large-particle method demonstrated a monotonic solution with a qualitative resolution of the details of the gas flow in the entire domain of determining the test problems. No spurious oscillations occurred during the process of fining the mesh, and convergence to the reference density profilewas observed. Theinfluenceof alimiter on thenumericaldissipation of theCDP2-UC schemeis analyzed. The results present the comparison with the following variants of the schemes: MUSCL(Monotone Upstream Scheme for ConservationLaws),MUSCL-CABARETwithaNOLD limiter(Non-OscillatoryLow-Dissipative),thediscontinuous Galerkin method with various forms of nonlinear correction, the hybrid weighted nonlinear scheme of the fourth order of approximation (CCSSR-HW4) and the popular WENO5 scheme (Weighted Essentially Non-Oscillatory Scheme) with fifth order of accuracy. The proposed algorithmsuccessfully competes with modern numerical methods that have a formally higher (fourth and fifth) order of approximation. The hybrid large-particle method has the simplicity, uniformity, and cost-effectiveness of the algorithm, as well as high resolution. The test calculations allowed the author to estimate the range of parametric control of the numerical dissipation of the method for correct numerical modeling of the applied problems with nonlinear wave fields and strong shock waves.