The Riemann problem for the Chaplygin gas dynamics with a single-point heating source

Abstract

This paper is devoted to the Riemann solutions to the Chaplygin gas dynamics with a single-point heating source. To deal with Dirac measure source term, we transform this model into a system of conservation laws and then present a definition of solution to this system of conservation laws. Based on this definition, we establish, respectively, the condition for the occurrence of stationary contact discontinuity solution and delta standing wave solution. With the help of these results, by analyzing the possible combination of various waves, ten kinds of exact solutions and the criteria for the emergence of each solution are established. It is observed that a delta standing wave solution arises for certain initial values, where two components of this solution contain the Dirac measure. It is also noticed that a stationary contact discontinuity followed by a delta shock wave emerges in some solutions. Moreover, these solutions can be used to test the validation of numerical algorithm for a system of conservation laws with a singular source term.