Delta Shock Waves as Flux-Approximation Limit of Solutions to the Modified Chaplygin Gas Equations

• DOI: 10.1007/s10440-019-00280-2
• Publication Date: 2019-07-19T08:04:48Z
• Authors: Jinjing Liu; Jie Liang; Hanchun Yang
• Keywords: Chaplygin gas

By introducing a triple parameter flux perturbation including pressure in the modified Chaplygin gas equations, we prove that, as the triple parameter flux perturbation vanishes, any two-shock Riemann solution and any two-rarefaction-wave Riemann solution to the perturbed modified Chaplygin gas equations tend to a delta-shock and a vacuum solution to the transport equations, respectively. Then we show that, as a double parameter flux perturbation vanishes, any two-shock Riemann solution under a certain condition to the perturbed modified Chaplygin gas equations tends to a delta shock wave solution to the generalized Chaplygin gas equations. In addition, we also show that, as a single parameter flux perturbation vanishes, any two-shock solution satisfying certain initial data and any parameterized delta shock wave solution to the perturbed generalized Chaplygin gas equations tend to a delta shock wave solution to the generalized Chaplygin gas equations. Finally, we exhibit some representative numerical simulations to confirm the theoretical analysis.