arXiv:2006.14971v1 [math.AP] 26 Jun 2020 Godunov type solvers for Hyperbolic Systems admitting δ − shocks Aekta Aggarwal 1 Ganesh Vaidya 2 G.D. Veerappa Gowda 2 1 Indian Institute of Management, Prabandh Shikhar, Rau–Pithampur Road, Indore, Madhya Pradesh 453556, India 2 TIFR Centre for Applicable Mathematics, Sharada Nagar, Near Yelahanka New Town Bus Stand, Chikkabommsandra, Bengaluru, Karnataka 560065 Abstract Godunov type numerical schemes for the class of hyperbolic sys- tems, admitting non-classical δ− shocks are proposed. It is shown that the numerical approximations converge to the solution and preserve the physical properties of the system such as positive density and bounded velocity. The scheme has been extended to positivity preserving and velocity bound pre- serving second-order accurate scheme by using appropriate slope limiters. The numerical results are compared with the existing the literature and the scheme is shown to capture the solution efficiently. The paper presents a hyperbolic system, for which an entropy satisfying scheme is constructed through an appropriate decoupling of the system into two scalar conserva- tion laws with discontinuous flux. Keyword: Discontinuous Flux, δ−shock, Transport Equation, Godunov Scheme, Generalized Pressureless Dynamics, Keyfitz Kranzer System 1 Introduction The following class of hyperbolic systems ρ t +(ρg(u)) x =0, (ρu) t +(ρug(u)+ P (ρ, u)) x = S (ρ) (1.1) finds numerous applications depending on the nature of the functions g, P, and S . These systems may admit non-classical shocks, namely δ- shock so- lutions. The investigation of delta shock waves has been increasingly active in the past over two decades. The delta shock wave is a generalization of a classical shock wave and is a kind of discontinuity, on which at least one of the state variables of the system (1.1) develops an extreme concentration in the form of a weighted Dirac delta function with the discontinuity as its 1