Asymptotic-numerical solution of nonlinear systems of one-dimensional balance laws

An asymptotic-numerical method is proposed to solve nonlinear scalar dissipative balance laws as well as systems of them in one space dimension, namely hyperbolic conservation laws affected by a certain kind of source term. Time asymptotics allows to obtain a hierarchy of coupled ordinary differential equations which can be solved by accurate methods. These provide first the long-time (stationary) solution, and then corrections to it to obtain an approximation valid at lower times. No accumulation of errors as time grows affects this method. On the contrary, results are more accurate at larger times. In the scalar case, an important role is played by the “auxiliary function” K ( u ) ≔ s ( u ) / f ′ ( u ) ′ , where f is the flux function and s is the source. A similar role is played by a certain matrix, in case of systems. Comparison is made with the Godunov method and with the AHOp (Asymptotic High-Order) numerical methods, recently developed by Natalini et al.