258 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
Lagrangian ADER-WENO Finite Volume Schemes on Unstructured Triangular Meshes Based On Genuinely Multidimensional HLL Riemann Solvers
In this paper we use the genuinely multidimensional HLL Riemann solvers
recently developed by Balsara et al. to construct a new class of
computationally efficient high order Lagrangian ADER-WENO one-step ALE finite
volume schemes on unstructured triangular meshes. A nonlinear WENO
reconstruction operator allows the algorithm to achieve high order of accuracy
in space, while high order of accuracy in time is obtained by the use of an
ADER time-stepping technique based on a local space-time Galerkin predictor.
The multidimensional HLL and HLLC Riemann solvers operate at each vertex of the
grid, considering the entire Voronoi neighborhood of each node and allows for
larger time steps than conventional one-dimensional Riemann solvers. The
results produced by the multidimensional Riemann solver are then used twice in
our one-step ALE algorithm: first, as a node solver that assigns a unique
velocity vector to each vertex, in order to preserve the continuity of the
computational mesh; second, as a building block for genuinely multidimensional
numerical flux evaluation that allows the scheme to run with larger time steps
compared to conventional finite volume schemes that use classical
one-dimensional Riemann solvers in normal direction. A rezoning step may be
necessary in order to overcome element overlapping or crossing-over. We apply
the method presented in this article to two systems of hyperbolic conservation
laws, namely the Euler equations of compressible gas dynamics and the equations
of ideal classical magneto-hydrodynamics (MHD). Convergence studies up to
fourth order of accuracy in space and time have been carried out. Several
numerical test problems have been solved to validate the new approach
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
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