(untitled) (Advances in Applied Mathematics and Mechanics, 2019)
This paper presents the understanding of the fundamentals when designing a numerical schemes for hyperbolic problems with discontinuities as parts of their solutions. The fundamentals include the consistency with hyperbolic balance laws in integral form rather than PDE form, spatial-temporal coupling, thermodynamic consistency for computing compressible fluid flows, convergence arguments and multidimensionality etc. Some numerical results are shown to display the performance.