1. Alves, M.A., Cruz, P., Mendes, A., Magalhães, F.D., Pinho, F.T., Oliveira, P.J.: Adaptive multiresolution approach for solution of hyperbolic PDEs. Comput. Methods Appl. Mech. Eng. 191, 3909-3928 (2002) 2. Bungartz, H.-J., Griebel, M.: Sparse grids. Acta Numerica 13, 147-269 (2004) 3. Capdeville, G.: A central WENO scheme for solving hyperbolic conservation laws on non-uniform meshes. J. Comput. Phys. 227, 2977-3014 (2008) 4. Carrillo, J.A., Gamba, I.M., Majorana, A., Shu, C.-W.: A WENO-solver for the transients of Boltzmann-Poisson system for semiconductor devices: performance and comparisons with Monte Carlo methods. J. Comput. Phys. 184, 498-525 (2003) 5. Castro, M., Costa, B., Don, W.S.: High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws. J. Comput. Phys. 230, 1766-1792 (2011) 6. Don, W.-S., Borges, R.: Accuracy of the weighted essentially non-oscillatory conservative finite difference schemes. J. Comput. Phys. 250, 347-372 (2013) 7. Dumbser, M., Käser, M., Titarev, V.A., Toro, E.F.: Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems. J. Comput. Phys. 226, 204-243 (2007) 8. Gottlieb, S., Shu, C.-W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43, 89-112 (2001) 9. Griebel, M., Schneider, M., Zenger, C.: A combination technique for the solution of sparse grid problems. In: Beauwens, R., de Groen, P. (eds) Iterative Methods in Linear Algebra, pp. 263-281. NorthHolland, Amsterdam (1992) 10. Guo, W., Cheng, Y.: A sparse grid discontinuous Galerkin method for high-dimensional transport equations and its application to kinetic simulations. SIAM J. Sci. Comput. 38, A3381-A3409 (2016) 11. Guo, W., Cheng, Y.: An adaptive multiresolution discontinuous Galerkin method for time-dependent transport equations in multidimensions. SIAM J. Sci. Comput. 39, A2962-A2992 (2017) 12. Henrick, A., Aslam, T., Powers, J.: Mapped weighted essentially non-oscillatory schemes: achieving optimal order near critical points. J. Comput. Phys. 207, 542-567 (2005) 13. Jiang, G.-S., Shu, C.-W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202-228 (1996) 14. Kolb, O.: A third order hierarchical basis WENO interpolation for sparse grids with application to conservation laws with uncertain data. J. Sci. Comput. 74, 1480-1503 (2018) 15. Lastdrager, B., Koren, B., Verwer, J.: Solution of time-dependent advection-diffusion problems with the sparse-grid combination technique and a rosenbrock solver. Comput. Methods Appl. Math. 1, 86-99 (2001) 16. Lastdrager, B., Koren, B., Verwer, J.: The sparse-grid combination technique applied to time-dependent advection problems. Appl. Numer. Math. 38, 377-401 (2001) 17. Levy, D., Puppo, G., Russo, G.: Central WENO schemes for hyperbolic systems of conservation laws. Math. Model. Numer. Anal. 33, 547-571 (1999) 18. Li, L., Zhu, J., Zhang, Y.-T.: Absolutely convergent fixed-point fast sweeping WENO methods for steady state of hyperbolic conservation laws. J. Comput. Phys. 443, 110516 (2021) 19. Liu, X.-D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115, 200-212 (1994) 20. Liu, Y., Cheng, Y., Shu, C.-W.: A simple bound-preserving sweeping technique for conservative numerical approximations. J. Sci. Comput. 73, 1028-1071 (2017) 21. Liu, Y., Zhang, Y.-T.: A robust reconstruction for unstructured WENO schemes. J. Sci. Comput. 54, 603-621 (2013) 22. Lu, D., Chen, S., Zhang, Y.-T.: Third order WENO scheme on sparse grids for hyperbolic equations. Pure Appl. Math. Q. 14, 57-86 (2018) 23. Lu, D., Zhang, Y.-T.: Krylov integration factor method on sparse grids for high spatial dimension convection-diffusion equations. J. Sci. Comput. 69, 736-763 (2016) 24. Qiu, J.-M., Christlieb, A.: A conservative high order semi-Lagrangian WENO method for the Vlasov equation. J. Comput. Phys. 229, 1130-1149 (2010) 25. Shu, C.-W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. In: Cockburn, B., Johnson, C., Shu, C.-W., Tadmor, E.(eds) Advanced Numerical Approximation of Nonlinear Hyperbolic Equations. Lecture Notes in Mathematics, Volume 1697. Springer (1998) 26. Tao, Z., Guo, W., Cheng, Y.: Sparse grid discontinuous Galerkin methods for the Vlasov-Maxwell system. J. Comput. Phys. X 3, 100022 (2019) 27. Yamaleev, N., Carpenter, M.: A systematic methodology for constructing high-order energy stable WENO schemes. J. Comput. Phys. 228, 4248-4272 (2009) 28. Zenger, C.: Sparse grids. In: Hackbusch, W. (ed) Notes on Numerical Fluid Mechanics, vol. 31, pp. 241-251. Vieweg, Braunschweig (1991) 29. Zhang, S., Jiang, S., Zhang, Y.-T., Shu, C.-W.: The mechanism of sound generation in the interaction between a shock wave and two counter rotating vortices. Phys. Fluids 21, 076101 (2009) 30. Zhang, Y.-T., Shu, C.-W.: High order WENO schemes for Hamilton-Jacobi equations on triangular meshes. SIAM J. Sci. Comput. 24, 1005-1030 (2003) 31. Zhang, Y.-T., Shu, C.-W.: Third order WENO scheme on three dimensional tetrahedral meshes. Commun. Comput. Phys. 5, 836-848 (2009) 32. Zhang, Y.-T., Shu, C.-W., Zhou, Y.: Effects of shock waves on Rayleigh-Taylor instability. Phys. Plasmas 13, 062705 (2006) 33. Zhu, J., Qiu, J.: A new fifth order finite difference WENO scheme for solving hyperbolic conservation laws. J. Comput. Phys. 318, 110-121 (2016) 34. Zhu, J., Shu, C.-W.: A new type of multi-resolution WENO schemes with increasingly higher order of accuracy. J. Comput. Phys. 375, 659-683 (2018) 35. Zhu, J., Shu, C.-W.: A new type of third-order finite volume multi-resolution WENO schemes on tetrahedral meshes. J. Comput. Phys. 406, 109212 (2020) 36. Zhu, J., Shu, C.-W.: Convergence to steady-state solutions of the new type of high-order multi-resolution WENO schemes: a numerical study. Commun. Appl. Math. Comput. 2, 429-460 (2020) 37. Zhu, X., Zhang, Y.-T.: Fast sparse grid simulations of fifth order WENO scheme for high dimensional hyperbolic PDEs. J. Sci. Comput. 87, 44 (2021) |