New computational techniques for finite-difference weighted essentially non-oscillatory schemes and related problems

Resumen

High-Resolution Shock-Capturing (HRSC) schemes constitute the state of the art for computing accurate numerical approximations to the solution of many hyperbolic systems of conservation laws, especially in computational fluid dynamics. A drawback of these schemes is that most of them use the spectral decomposition of the Jacobian matrix of the system to compute the numerical approximations by local projections to characteristic fields. The numerical solutions obtained are often excellent in terms of resolution, but the computational effort needed may be too high for some problems, especially those for which the spectral information of the flux Jacobian matrix is not available or is quite difficult to obtain. In order to reduce the computational cost, we can use component-wise finite-difference WENO schemes, based on Shu-Osher's finite-difference schemes, which compute the numerical fluxes at each cell interface by upwind-biased reconstructions of split upwind fluxes, avoiding the use of the characteristic information, but, unfortunately, they tend to yield results that are too diffusive and oscillatory. In this work we develop some techniques to improve the accuracy of the numerical results obtained with finite-difference WENO schemes, but also the efficiency of those schemes. There are a lot of works that analyze the main parts of WENO schemes, as the definition of the weights, the smoothness indicators or the role of some parameters present in the definition of the weights in the loss of accuracy near discontinuities and extrema. We derive new weights for the WENO scheme and get some constraints on those parameters present in their definition to guarantee maximal order for sufficiently smooth solutions with an arbitrary number of vanishing derivatives. The other basic ingredient of WENO finite-difference schemes is the use of the upwinding when computing the numerical flux function. The sophisticated design of the numerical flux function, that incorporates upwinding through characteristic information that needs to be computed at each cell boundary in the computational domain, tends to be fairly expensive. To speed up computing times, we use component-wise schemes that avoid the use of characteristic information when computing the numerical fluxes. We introduce an alternative flux-splitting to the usual Lax-Friedrichs flux-splitting. The use of this flux-splitting leads to more accurate numerical solutions, especially near discontinuities, where the use of this flux-splitting reduces the dissipation of the numerical solutions. In the case of the numerical simulation of shallow water flows it has been studied that to accurately represent discontinuous behavior, known to occur due to the non-linear hyperbolic nature of the shallow water system, and, at the same time, numerically maintain stationary solutions it is necessary the use of well-balanced shock-capturing (WBSC) schemes. In this work we combine the block structured AMR technique with a well-balanced scheme to develop a combined AMR-WBSC scheme. We show that in order for the combined AMR-WBSC scheme to maintain its well-balanced character it is necessary to implement well-balanced interpolatory techniques in the transfer operators involved in the multi-level structure. It is shown that the new AMR-WBSC scheme is more efficient than usual WBSC schemes and that it preserves the "water at rest" stationary solutions as the underlying WBSC does. We make extensive testing to compare the performance of several schemes and support our discussion.