376,204 research outputs found
Positivity-Preserving Finite Difference WENO Schemes with Constrained Transport for Ideal Magnetohydrodynamic Equations
In this paper, we utilize the maximum-principle-preserving flux limiting
technique, originally designed for high order weighted essentially
non-oscillatory (WENO) methods for scalar hyperbolic conservation laws, to
develop a class of high order positivity-preserving finite difference WENO
methods for the ideal magnetohydrodynamic (MHD) equations. Our schemes, under
the constrained transport (CT) framework, can achieve high order accuracy, a
discrete divergence-free condition and positivity of the numerical solution
simultaneously. Numerical examples in 1D, 2D and 3D are provided to demonstrate
the performance of the proposed method.Comment: 21 pages, 28 figure
A posteriori analysis of fully discrete method of lines DG schemes for systems of conservation laws
We present reliable a posteriori estimators for some fully discrete schemes
applied to nonlinear systems of hyperbolic conservation laws in one space
dimension with strictly convex entropy. The schemes are based on a method of
lines approach combining discontinuous Galerkin spatial discretization with
single- or multi-step methods in time. The construction of the estimators
requires a reconstruction in time for which we present a very general framework
first for odes and then apply the approach to conservation laws. The
reconstruction does not depend on the actual method used for evolving the
solution in time. Most importantly it covers in addition to implicit methods
also the wide range of explicit methods typically used to solve conservation
laws. For the spatial discretization, we allow for standard choices of
numerical fluxes. We use reconstructions of the discrete solution together with
the relative entropy stability framework, which leads to error control in the
case of smooth solutions. We study under which conditions on the numerical flux
the estimate is of optimal order pre-shock. While the estimator we derive is
computable and valid post-shock for fixed meshsize, it will blow up as the
meshsize tends to zero. This is due to a breakdown of the relative entropy
framework when discontinuities develop. We conclude with some numerical
benchmarking to test the robustness of the derived estimator
Self-adaptive moving mesh schemes for short pulse type equations and their Lax pairs
Integrable self-adaptive moving mesh schemes for short pulse type equations
(the short pulse equation, the coupled short pulse equation, and the complex
short pulse equation) are investigated. Two systematic methods, one is based on
bilinear equations and another is based on Lax pairs, are shown. Self-adaptive
moving mesh schemes consist of two semi-discrete equations in which the time is
continuous and the space is discrete. In self-adaptive moving mesh schemes, one
of two equations is an evolution equation of mesh intervals which is deeply
related to a discrete analogue of a reciprocal (hodograph) transformation. An
evolution equations of mesh intervals is a discrete analogue of a conservation
law of an original equation, and a set of mesh intervals corresponds to a
conserved density which play an important role in generation of adaptive moving
mesh. Lax pairs of self-adaptive moving mesh schemes for short pulse type
equations are obtained by discretization of Lax pairs of short pulse type
equations, thus the existence of Lax pairs guarantees the integrability of
self-adaptive moving mesh schemes for short pulse type equations. It is also
shown that self-adaptive moving mesh schemes for short pulse type equations
provide good numerical results by using standard time-marching methods such as
the improved Euler's method.Comment: 13 pages, 6 figures, To be appeared in Journal of Math-for-Industr
Invariant Discretization Schemes Using Evolution-Projection Techniques
Finite difference discretization schemes preserving a subgroup of the maximal
Lie invariance group of the one-dimensional linear heat equation are
determined. These invariant schemes are constructed using the invariantization
procedure for non-invariant schemes of the heat equation in computational
coordinates. We propose a new methodology for handling moving discretization
grids which are generally indispensable for invariant numerical schemes. The
idea is to use the invariant grid equation, which determines the locations of
the grid point at the next time level only for a single integration step and
then to project the obtained solution to the regular grid using invariant
interpolation schemes. This guarantees that the scheme is invariant and allows
one to work on the simpler stationary grids. The discretization errors of the
invariant schemes are established and their convergence rates are estimated.
Numerical tests are carried out to shed some light on the numerical properties
of invariant discretization schemes using the proposed evolution-projection
strategy
- …