Hybrid Weighted Essentially Non-oscillatory Schemes With Different Indicators

The weighted essentially non-oscillatory (WENO) schemes for solving nonlinear hyperbolic conservation laws and convection dominated problems are high-order accu-rate and maintain stable, essentially non-oscillatory and sharp discontinuity transition. The schemes are thus especially suitable for problems containing both strong disconti-nuities and complex smooth solution features. WENO schemes are robust and do not require the user to tune parameters. With the above properties, WENO schemes become the key numerical method in the computational fluid dynamics domain, and have been generalized to the following applications in areas including computational astronomy and astrophysics, semiconductor device simulation, traffic flow models, computational biology. A key idea in finite difference WENO schemes is a combination of lower order fluxes on candidate stencil to obtain a higher order approximation. The choice of the weight to each candidate stencil, which is a nonlinear function of the grid values, is cru-cial to the success of WENO schemes. For the system case, WENO schemes are based on local characteristic decompositions and flux splitting to avoid spurious numerical oscillations. But the cost of computation of nonlinear weights and local characteristic decompositions is very high. With the increase of the number of equation and space dimension, the trend of the increase of computational cost is more evident. This draw-back makes it necessary to implement the research for the efficient WENO schemes. At the same time, there have been little efforts in the literature to overcome the drawback, so we carry out the work in this dissertation.We focus our research on the hybrid WENO schemes to hybridize the WENO reconstruction (using nonlinear weights) and the simple up-wind linear reconstruction (using linear weights). The key idea is to use the WENO reconstruction in the dis-continuous regions to capture discontinuities, and to use the efficient up-wind linear reconstruction in order to avoid the usage of nonlinear weight and the local character-istic decomposition procedure. Our aim is to apply the hybrid WENO schemes to save computational cost considerably and at the same time to maintain the original good properties of the WENO scheme. Compared with the other hybrid WENO schemes, the hybrid WENO scheme of this paper has the following properties:its key idea is more direct than others, due to the close relation between the WENO reconstruction and the up-wind linear reconstruction; the accuracy of the two type reconstructions is consistent, so the numerical fluxes are smoother. An important component of the hy-brid scheme is a "troubled-cell" (cell contains discontinuity) indicator to automatically identify where the discontinuity of the solution is. In this paper, we reconstruct the troubled-cell indicators mainly based on the limiters from the discontinuous Galerkin (DG) methods. By extensive numerical experiments, we design and choose practical troubled-cell indicators for the hybrid WENO schemes, such that the hybrid WENO schemes with the indicators have higher efficiency than the original WENO schemes and maintain the good properties of the original WENO schemes at the same time and have practical value.Firstly, for the one-dimensional compressible Euler equations, we test the hybrid WENO schemes with none indicators by benchmark numerical examples. We find that ATV, TVB, MR and KXRCF indicators are better than others. For they result in little CPU time, more accurate numerical solutions and smaller percentages of fluxes by WENO reconstruction than the remaining indicators.Subsequently, we extend the hybrid WENO schemes with the above four indi-cators to Burgers equation, multi-dimensional Euler equations. The numerical results indicate that the hybrid WENO schemes can save computational cost considerabley.For the hyperbolic conservation law equation with source terms:shallow water equations and pollutant transport equations in shallow water, we design hybrid well-balanced WENO schemes with indicators. Theoretical analysis and numerical test ver-ify that the hybrid well-balanced WENO schemes can maintain the exact conservation property (exact C-property). The property is important for the balance between the flux gradient and the source terms.In order to handle the general physical domain well, we decompose the physi- cal domain into curvilinear grids. A given coordinate transformation maps the general physical domain to a Cartesian computational domain and maps the curvilinear grid in physical domain to a regular grid in computational domain. Meanwhile, we trans-form the governing equation in physical space to computational space. At last, we generalize the hybrid WENO schemes to the hyperbolic conservation law equation in computational space. The numerical results are desirable.Extensive numerical examples based on multi-dimensional, on different type prob-lems and on different type grids suggest that the hybrid WENO schemes with indica-tors can save computational cost considerably and maintain the property of the original WENO schemes at the same time.In summary, the hybrid WENO schemes with the ATV, TVB, MR and KXRCF indicators is efficiency effective and robust.