Improved Weighted Essentially Non-oscillatory Scheme is a high order finite volume method. The mixed stencils can be obtained by a combination of r + 1 order and r order stencils. We improve the weights by the mapping method. The restriction that conventional ENO or WENO schemes only use r order stencils, is removed. Higher resolution can be achieved by introducing the r + 1 order stencils. This method is verified by three cases, i.e. the interaction of a moving shock with a density wave problem, the interacting blast wave problem and the double mach reflection problem. The numerical results show that the Improved Weighted Essential Non-oscillatory method is a stable, accurate high-resolution finite volume scheme.
References
[1]
Harten, A. (1986) On High-Order Accurate Interpolation for Non-Oscillatory Shock Capturing Schemes. In: Dafermos, C., et al., Eds., Oscillation Theory. Computation and Methods of Compensated Compactness, Springer-Verlag, New York, 71-105. https://doi.org/10.1007/978-1-4613-8689-6_4
[2]
Harten, A. and Osher, S. (1987) Uniformly High Order Accurate Essential Non-Oscillatory Schemes, I. SIAM Journal on Numerical Analysis, 24, 279-309. https://doi.org/10.1137/0724022
[3]
Liu, X., Osher, S. and Chan, T. (1994) Weighted Essentially Non-Oscillatory Schemes. Journal of Computational Physics, 115, 200-212. https://doi.org/10.1006/jcph.1994.1187
[4]
Jiang, G.-S. and Shu, C.-W. (1996) Efficient Implementation of Weighted ENO Schemes. Journal of Computational Physics, 126, 202-228. https://doi.org/10.1006/jcph.1996.0130
[5]
Bryson, S., and Levy, D. (2003) High-Order Semi-Discrete Central-Upwind Schemes for Multi-Dimensional Hamilton-Jacobi Equations. Journal of Computational Physics, 189, 63-87. https://doi.org/10.1016/S0021-9991(03)00201-8
[6]
Kim, D. and Kwon, J.H. (2005) A High-Order Accurate Hybrid Scheme Using a Central Flux Scheme and a WENO Scheme for Compressible Flowfield Analysis. Journal of Computational Physics, 210, 554-583. https://doi.org/10.1016/j.jcp.2005.04.023
[7]
Johnsen, E. and Colonius, T. (2006) Implementation of WENO Schemes in Compressible Multicomponent Flow Problems. Journal of Computational Physics, 219, 715-732. https://doi.org/10.1016/j.jcp.2006.04.018
[8]
Caleffi, V., Valiani, A. and Bernin, A. (2006) Fourth-Order Balanced Source Term Treatment in Central WENO Schemes for Shallow Water Equations. Journal of Computational Physics, 218, 228-245. https://doi.org/10.1016/j.jcp.2006.02.001
[9]
Henrick, A.K., Aslam, T.D. and Powers, J.M. (2005) Mapped Weighted Essentially Non-Oscillatory Schemes: Achieving Optimal Order near Critical Points. Journal of Computational Physics, 207, 542-567. https://doi.org/10.1016/j.jcp.2005.01.023
[10]
Martin, M.P., Taylor, E.M., Wu, M. and Weirs, V.G. (2006) A Bandwidth-Optimized WENO Scheme for the Effective Direct Numerical Simulation of Compressible Turbulence. Journal of Computational Physics, 220, 270-289. https://doi.org/10.1016/j.jcp.2006.05.009
[11]
Zheng, F., Shu, C.W. and Qiu, J. (2019) High Order Finite Difference Hermite WENO Schemes for the Hamilton-Jacobi Equations on Unstructured Meshes. Computers & Fluids, 183, 53-65. https://doi.org/10.1016/j.compfluid.2019.02.010
[12]
Wu, L., Zhang, Y.T., Zhang, S., et al. (2016) High Order Fixed-Point Sweeping WENO Methods for Steady State of Hyperbolic Conservation Laws and Its Convergence Study. Communications in Computational Physics, 20, 835-869. https://doi.org/10.4208/cicp.130715.010216a
[13]
Shu, C.-W. (1997) Essential Non-Oscillatory and Weighted Essential Non-Oscillatory Schemes for Hyperbolic Conservation Laws. NASA CR-97-206253. https://doi.org/10.1007/BFb0096355
[14]
Shu, C.-W. (1997) Preface to the Republication of “Uniformly High Order Essentially Non-Oscillatory Schemes, III,” by Harten, Engquist, Osher, and Chakravarthy. Journal of Computational Physics, 131, 1-2. https://doi.org/10.1006/jcph.1996.5630
Zong, W.G., Deng, X.G. and Zhang, H.X. (2003) Implementation of DWENO Schemes and Its Application. Acta Aerodynamica Sinica, 21, 399-407.
[17]
Jun, Z. and Ning, Z. (2005) A Kind of MWENO Schemes and Its Application. Acta Aerodynamica Sinica, 23, 330-336.
[18]
Rusanov, V.V. (1961) Calculation of Interaction of Non-Steady Shock Waves with Obstacles. National Research Council of Canada, Ottawa.
[19]
Levy, D., Puppo, G. and Russo, G. (1999) Central WENO Schemes for Hyperbolic Systems of Conservation Laws. Mathematical Modelling and Numerical Analysis, 33, 547-571. https://doi.org/10.1051/m2an:1999152
[20]
Levy, D., Puppo, G. and Russo, G. (2002) A Fourth-Order Central WENO Scheme for Multidimensional Hyperbolic Systems of Conservation Laws. SIAM Journal on Scientific Computing, 24, 480-506. https://doi.org/10.1137/S1064827501385852
[21]
Shu, C. (1998) Essentialy Non-Oscilatory and Weighted Essential Non-Oscilatory Schemes for Hyperbolic Conservation Laws. In: Ockburn, B., Jo Hnsonc, S., Hu, C. and Tadmor, E., Eds., Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Lecture Notes in Mathematics, Vol. 1697, Springer-Verlag, Berlin, 325-432. https://doi.org/10.1007/BFb0096355
[22]
Shu, C.W. and Osher, S. (1998) Efficient Implementation of Essentially Non-Oscillatory Shock Capturing Schemes II. Journal of Computational Physics, 83, 32-78. https://doi.org/10.1016/0021-9991(89)90222-2
[23]
Woodward, P. and Colella, P. (1984) The Numerical Simulation of Two-Dimensional Fluid Flow with Strong Shocks. Journal of Computational Physics, 54, 115-173. https://doi.org/10.1016/0021-9991(84)90142-6
