Radially Symmetrical Problems for Compressible Fluids with a High-Resolution Boundary Condition

Abstract

Imposing appropriate numerical boundary conditions at the symmetrical center $r=0$ is vital when computing compressible fluids with radial symmetry. Extrapolation and other traditional techniques are often employed, but spurious numerical oscillations or wall-heating phenomena can occur. In this paper, we emphasize that because of the conservation property, the updating formula of the boundary cell average can coincide with the one for interior cell averages. To achieve second-order accuracy both in time and space, we associate obtaining the inner boundary value at $r=0$ with the resolution of the corresponding one-sided generalized Riemann problem (GRP). Acoustic approximation is applied in this process. It creates conditions to avoid the singularity of type $1/r$ and aids in obtaining the value of the singular quantity using L'Hospital's rule. Several challenging scenarios are tested to demonstrate the effectiveness and robustness of our approach.