Efficient implementation of essentially non-oscillatory shock-capturing schemes
Journal of Computational Physics
Multidimensional upwind methods for hyperbolic conservation laws
Journal of Computational Physics
On Godunov-type methods near low densities
Journal of Computational Physics
Weighted essentially non-oscillatory schemes
Journal of Computational Physics
Efficient implementation of weighted ENO schemes
Journal of Computational Physics
Wave propagation algorithms for multidimensional hyperbolic systems
Journal of Computational Physics
The Runge-Kutta discontinuous Galerkin method for conservation laws V multidimensional systems
Journal of Computational Physics
Journal of Computational Physics
Maintaining pressure positivity in magnetohydrodynamic simulations
Journal of Computational Physics
Journal of Computational Physics
A New Class of Optimal High-Order Strong-Stability-Preserving Time Discretization Methods
SIAM Journal on Numerical Analysis
ADER: Arbitrary High Order Godunov Approach
Journal of Scientific Computing
Non-linear evolution using optimal fourth-order strong-stability-preserving Runge-Kutta methods
Mathematics and Computers in Simulation - Nonlinear waves: computation and theory II
Finite-volume WENO schemes for three-dimensional conservation laws
Journal of Computational Physics
Numerical Simulation of High Mach Number Astrophysical Jets with Radiative Cooling
Journal of Scientific Computing
A multi-state HLL approximate Riemann solver for ideal magnetohydrodynamics
Journal of Computational Physics
Journal of Computational Physics
Positive Scheme Numerical Simulation of High Mach Number Astrophysical Jets
Journal of Scientific Computing
Journal of Computational Physics
Journal of Computational Physics
Hierarchical reconstruction for spectral volume method on unstructured grids
Journal of Computational Physics
A positive MUSCL-Hancock scheme for ideal magnetohydrodynamics
Journal of Computational Physics
Efficient implementation of essentially non-oscillatory shock-capturing schemes, II
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
Positivity-preserving DG and central DG methods for ideal MHD equations
Journal of Computational Physics
A HLL-Rankine-Hugoniot Riemann solver for complex non-linear hyperbolic problems
Journal of Computational Physics
High-order solution-adaptive central essentially non-oscillatory (CENO) method for viscous flows
Journal of Computational Physics
Journal of Computational Physics
Hi-index | 31.47 |
Several computational problems in science and engineering are stringent enough that maintaining positivity of density and pressure can become a problem. We build on the realization that positivity can be lost within a zone when reconstruction is carried out in the zone. We present a multidimensional, self-adjusting strategy for enforcing the positivity of density and pressure in hydrodynamic and magnetohydrodynamic (MHD) simulations. The MHD case has never been addressed before, and the hydrodynamic case has never been presented in quite the same way as done here. The method examines the local flow to identify regions with strong shocks. The permitted range of densities and pressures is also obtained at each zone by examining neighboring zones. The range is expanded if the solution is free of strong shocks in order to accommodate higher order non-oscillatory reconstructions. The density and pressure are then brought into the permitted range. The method has also been extended to MHD. It is very efficient and should extend to discontinuous Galerkin methods as well as flows on unstructured meshes.Video 1 The method presented here does not degrade the order of accuracy for smooth flows. Via a stringent test suite, we document that our method works well on structured meshes for all orders of accuracy up to four. When the same test problems are run without the positivity preserving methods, one sees a very clear degradation in the results, highlighting the value of the present method. The results are compelling because realistic simulation of several difficult astrophysical and space physics problems requires the use of parameters that are similar to the ones in our test problems. In this work, weighted non-oscillatory reconstruction was applied to the conserved variables, i.e. we did not apply the reconstruction to the characteristic variables, which would have made the scheme more expensive. Yet, used in conjunction with the positivity preserving schemes presented here, the less expensive reconstruction works very well in two and three dimensions. This suggests that when designing robust, high accuracy schemes, having a self-adjusting positivity criterion is almost as important as the non-linear hybridization.