This is the continuation of the paper”Central discontinuous Galerkin methods on overlapping cells with a non-oscillatory hierarchical reconstruction”by the same authors.The hierarchical reconstruction introduced the...This is the continuation of the paper”Central discontinuous Galerkin methods on overlapping cells with a non-oscillatory hierarchical reconstruction”by the same authors.The hierarchical reconstruction introduced therein is applied to central schemes on overlapping cells and to finite volume schemes on non-staggered grids.This takes a new finite volume approach for approximating non-smooth solutions.A critical step for high-order finite volume schemes is to reconstruct a non-oscillatory high degree polynomial approximation in each cell out of nearby cell averages.In the paper this procedure is accomplished in two steps:first to reconstruct a high degree polynomial in each cell by using e.g.,a central reconstruction,which is easy to do despite the fact that the reconstructed polynomial could be oscillatory;then to apply the hierarchical reconstruction to remove the spurious oscillations while maintaining the high resolution.All numerical computations for systems of conservation laws are performed without characteristic decomposition.In particular,we demonstrate that this new approach can generate essentially non-oscillatory solutions even for 5th-order schemes without characteristic decomposition.展开更多
We consider constraint preserving multidimensional evolution equations.A prototypical example is provided by the magnetic induction equation of plasma physics.The constraint of interest is the divergence of the magnet...We consider constraint preserving multidimensional evolution equations.A prototypical example is provided by the magnetic induction equation of plasma physics.The constraint of interest is the divergence of the magnetic field.We design finite volume schemes which approximate these equations in a stable manner and preserve a discrete version of the constraint.The schemes are based on reformulating standard edge centered finite volume fluxes in terms of vertex centered potentials.The potential-based approach provides a general framework for faithful discretizations of constraint transport and we apply it to both divergence preserving as well as curl preserving equations.We present benchmark numerical tests which confirm that our potential-based schemes achieve high resolution,while being constraint preserving.展开更多
基金supported in part by NSF grant DMS-0511815.The research of C.-W.Shu was supported in part by the Chinese Academy of Sciences while this author was visiting the University of Science and Technology of China(grant 2004-1-8)+3 种基金the Institute of Computational Mathematics and Scientific/Engineering ComputingAdditional support was provided by ARO grant W911NF-04-1-0291 and NSF grant DMS0510345The research of E.Tadmor was supported in part by NSF grant 04-07704 and ONR grant N00014-91-J-1076The research of M.Zhang was supported in part by the Chinese Academy of Sciences grant 2004-1-8.
文摘This is the continuation of the paper”Central discontinuous Galerkin methods on overlapping cells with a non-oscillatory hierarchical reconstruction”by the same authors.The hierarchical reconstruction introduced therein is applied to central schemes on overlapping cells and to finite volume schemes on non-staggered grids.This takes a new finite volume approach for approximating non-smooth solutions.A critical step for high-order finite volume schemes is to reconstruct a non-oscillatory high degree polynomial approximation in each cell out of nearby cell averages.In the paper this procedure is accomplished in two steps:first to reconstruct a high degree polynomial in each cell by using e.g.,a central reconstruction,which is easy to do despite the fact that the reconstructed polynomial could be oscillatory;then to apply the hierarchical reconstruction to remove the spurious oscillations while maintaining the high resolution.All numerical computations for systems of conservation laws are performed without characteristic decomposition.In particular,we demonstrate that this new approach can generate essentially non-oscillatory solutions even for 5th-order schemes without characteristic decomposition.
基金E.Tadmor Research was supported in part by NSF grant 07-07949 and ONR grant N00014-091-0385.
文摘We consider constraint preserving multidimensional evolution equations.A prototypical example is provided by the magnetic induction equation of plasma physics.The constraint of interest is the divergence of the magnetic field.We design finite volume schemes which approximate these equations in a stable manner and preserve a discrete version of the constraint.The schemes are based on reformulating standard edge centered finite volume fluxes in terms of vertex centered potentials.The potential-based approach provides a general framework for faithful discretizations of constraint transport and we apply it to both divergence preserving as well as curl preserving equations.We present benchmark numerical tests which confirm that our potential-based schemes achieve high resolution,while being constraint preserving.
