10 research outputs found
An explicit high-order single-stage single-step positivity-preserving finite difference WENO method for the compressible Euler equations
In this work we construct a high-order, single-stage, single-step
positivity-preserving method for the compressible Euler equations. Space is
discretized with the finite difference weighted essentially non-oscillatory
(WENO) method. Time is discretized through a Lax-Wendroff procedure that is
constructed from the Picard integral formulation (PIF) of the partial
differential equation. The method can be viewed as a modified flux approach,
where a linear combination of a low- and high-order flux defines the numerical
flux used for a single-step update. The coefficients of the linear combination
are constructed by solving a simple optimization problem at each time step. The
high-order flux itself is constructed through the use of Taylor series and the
Cauchy-Kowalewski procedure that incorporates higher-order terms. Numerical
results in one- and two-dimensions are presented
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
A general positivity-preserving algorithm for implicit high-order finite volume schemes solving the Euler and Navier-Stokes equations
This paper presents a general positivity-preserving algorithm for implicit
high-order finite volume schemes solving Euler and Navier-Stokes equations.
Previous positivity-preserving algorithms are mainly based on mathematical
analyses, being highly dependent on the existence of low-order
positivity-preserving numerical schemes for specific governing equations. This
dependency poses serious restrictions on extending these algorithms to
temporally implicit schemes, since it is difficult to know if a low-order
implicit scheme is positivity-preserving. The present positivity-preserving
algorithm is based on an asymptotic analysis of the solutions near local vacuum
minimum points. The asymptotic analysis shows that the solutions decay
exponentially with time to maintain non-negative density and pressure at a
local vacuum minimum point. In its neighborhood, the exponential evolution
leads to a modification of the linear evolution process, which can be modelled
by a direct correction of the linear residual to ensure positivity. This
correction however destroys the conservation of the numerical scheme.
Therefore, a second correction procedure is proposed to recover conservation.
The proposed positivity-preserving algorithm is considerably less restrictive
than existing algorithms. It does not rely on the existence of low-order
positivity-preserving baseline schemes and the convex decomposition of volume
integrals of flow quantities. It does not need to reduce the time step size for
maintaining the stability either. Furthermore, it can be implemented
iteratively in the implicit dual time-stepping schemes to preserve positivity
of the intermediate and converged states of the sub-iterations. It is proved
that the present positivity-preserving algorithm is accuracy-preserving.
Numerical results demonstrate that the proposed algorithm preserves the
positive density and pressure in all test cases.Comment: 52 pages, 8 figure