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WENO schemes with Marquina's flux splitting

The commonly used types of finite difference WENO schemes use the Roe with entropy fix and locally Lax-Friedrichs flux splitting. In both cases the flux values at the cell boundaries are obtained by using the WENO reconstruction of the local characteristic variables. Typically, the characteristic fields are determined from the Jacobian matrix of the flux evaluated at some reasonable average of the adjacent states, Roe or van Leer average for example. On the other hand Marquina introduced a flux splitting technique based on the characteristic field values obtained from the Jacobian matrixes computed separately at the left and the right states of the considered cell boundary. In this paper the Marquina's flux is used in combination with the finite difference WENO schemes. Moreover, we extend the considered schemes to the balance laws. This extension is made such that the balancing between the flux gradient and the source term is achieved. The comparison with other finite difference WENO schemes is made through several shallow water and open-channel flow test problems.

Weno; Marquina's flux; balance laws; shallow water; open-channel

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