65 research outputs found
High Order Maximum Principle Preserving Semi-Lagrangian Finite Difference WENO schemes for the Vlasov Equation
In this paper, we propose the parametrized maximum principle preserving (MPP)
flux limiter, originally developed in [Z. Xu, Math. Comp., (2013), in press],
to the semi- Lagrangian finite difference weighted essentially non-oscillatory
scheme for solving the Vlasov equation. The MPP flux limiter is proved to
maintain up to fourth order accuracy for the semi-Lagrangian finite difference
scheme without any time step restriction. Numerical studies on the
Vlasov-Poisson system demonstrate the performance of the proposed method and
its ability in preserving the positivity of the probability distribution
function while maintaining the high order accuracy
High-order accurate physical-constraints-preserving finite difference WENO schemes for special relativistic hydrodynamics
The paper develops high-order accurate physical-constraints-preserving finite
difference WENO schemes for special relativistic hydrodynamical (RHD)
equations, built on the local Lax-Friedrich splitting, the WENO reconstruction,
the physical-constraints-preserving flux limiter, and the high-order strong
stability preserving time discretization. They are extensions of the
positivity-preserving finite difference WENO schemes for the non-relativistic
Euler equations. However, developing physical-constraints-preserving methods
for the RHD system becomes much more difficult than the non-relativistic case
because of the strongly coupling between the RHD equations, no explicit
expressions of the primitive variables and the flux vectors, in terms of the
conservative vector, and one more physical constraint for the fluid velocity in
addition to the positivity of the rest-mass density and the pressure. The key
is to prove the convexity and other properties of the admissible state set and
discover a concave function with respect to the conservative vector replacing
the pressure which is an important ingredient to enforce the
positivity-preserving property for the non-relativistic case. Several one- and
two-dimensional numerical examples are used to demonstrate accuracy,
robustness, and effectiveness of the proposed physical-constraints-preserving
schemes in solving RHD problems with large Lorentz factor, or strong
discontinuities, or low rest-mass density or pressure etc.Comment: 39 pages, 13 figure
High-Resolution Mathematical and Numerical Analysis of Involution-Constrained PDEs
Partial differential equations constrained by involutions provide the highest fidelity mathematical models for a large number of complex physical systems of fundamental interest in critical scientific and technological disciplines. The applications described by these models include electromagnetics, continuum dynamics of solid media, and general relativity. This workshop brought together pure and applied mathematicians to discuss current research that cuts across these various disciplines’ boundaries. The presented material illuminated fundamental issues as well as evolving theoretical and algorithmic approaches for PDEs with involutions. The scope of the material covered was broad, and the discussions conducted during the workshop were lively and far-reaching
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