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Analysis of Nonlinear Resonance in Conservation Laws with Point Sources and Well‐Balanced Scheme

The purpose of this paper is to study the wave behavior of the hyperbolic conservation law with concatenation of point sources: for i∈I some finite index set, and where δ( ) is the Dirac measure. Special features of this problem are the discontinuities that appear along the t‐axis at the point sources x=xi+1/2. Resonance occurs when the speed of the nonlinear wave is close to zero. In addition to classical shock waves, the equation exhibits overcompressive and undercompressive waves. The Riemann problem “with resonance” is solved, and we show global existence via the Glimm scheme. Analytical understanding is used to design a well‐balanced numerical scheme, of the Godunov type, which preserves the balance between the sources terms and the fluxes terms. Some numerical tests are reported.