First- and Second-Order Flux Difference Splitting Schemes for Dam-Break Problem

The first-order flux difference splitting scheme and its second-order extensions are investigated for their applicability to dam-break problems. Roe's first-order explicit scheme is first formulated using an approximate Jacobian. A general entropy-satisfying formula is incorporated, which significantly improves the applicability of the Roe scheme. The Roe scheme is extended to second-order accuracy using the Lax-Wendroff numerical flux, the MUSCL approach, and the modified flux approach. To damp out oscillations resulting from the second order of accuracy, a flux/slope limiter is incorporated in the second-order schemes. Numerical results for dam-break problems demonstrating the effect of the violation of the entropy-inequality condition and effectiveness of the proposed treatment by a general entropy-satisfying formula are presented. The Roe scheme is compared against its second-order extensions as well as with first-order schemes such as the Lax-Friedrichs and modified Beam and Warming schemes. It is demonstrated that although higher-order schemes provide better shock resolution, Roe's first-order scheme may be preferred for practical applications when computation time, overall accuracy, and applicability are considered.