Development of Numerical Program on 2-Dimensional Inviscid Burgers Equation in Finite Volume Method with UPWIND and TVD Scheme

Abstract

This paper presents the development and validation of a numerical solver for the two-dimensional steady-state inviscid Burgers equation using the Finite Volume Method (FVM). The Burgers equation serves as a fundamental model for studying nonlinear wave propagation and shock formation in hyperbolic conservation laws. Two numerical schemes are implemented and compared: the first-order Upwind scheme and high-resolution Total Variation Diminishing (TVD) schemes with various flux limiters including Min-Mod, Van Leer, Superbee, QUICK, and UMIST. The numerical solutions are validated against an exact analytical solution derived using the method of characteristics. Results demonstrate that while the Upwind scheme is robust and stable, it exhibits significant numerical diffusion that smears shock discontinuities. In contrast, TVD schemes achieve sharper shock resolution while maintaining monotonicity. The upwind scheme spreads the shock over 3.86 cells, whereas the limiters confine it to between 1.73 and 1.91 cells, reduce the L1 error by 60 to 67 per cent and reproduce the total variation of the exact solution exactly. Superbee gives the sharpest shock and QUICK the lowest L2 error, while Min-Mod is the most diffusive of the limiters; the most compressive limiters, however, only reach a steady state once the CFL number is reduced. A refinement study over nine grids gives observed orders of accuracy of 0.86 in L1 and 0.52 in L2 with a non-convergent maximum error, which is the behaviour expected of a first-order monotone scheme applied to a discontinuous solution.