A high-resolution scheme for axisymmetric hydrodynamics based on the 2D GRP solvers

Abstract

In the simulation of axisymmetric fluids, imposing an appropriate numerical boundary condition at the symmetrical axis r=0 is an inevitable difficulty. To maintaining the conservation property, we choose to keep its form coincided with the numerical scheme on interior control volumes. Then we obtain necessary interface values in both scenarios from solving the two-dimensional generalized Riemann problems (2D GRPs) and their one-sided versions. Therein the effects of transversal variation and geometrical source are specifically emphasized to exhibit the genuine multi-dimensionality of the entire algorithm. In the construction of the one-sided 2D GRP solver, we have to face the difficulty brought by the singularity of the source at r=0. The ingenious combination of acoustic approximation, symmetry argument, and L’Hospital’s rule is proposed in obtaining a pivotal limiting value there. Several challenging tests are provided to demonstrate the effectiveness and robustness of our approach.