105 research outputs found
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
Second-order accurate genuine BGK schemes for the ultra-relativistic flow simulations
This paper presents second-order accurate genuine BGK (Bhatnagar-Gross-Krook)
schemes in the framework of finite volume method for the ultra-relativistic
flows. Different from the existing kinetic flux-vector splitting (KFVS) or
BGK-type schemes for the ultra-relativistic Euler equations, the present
genuine BGK schemes are derived from the analytical solution of the
Anderson-Witting model, which is given for the first time and includes the
"genuine" particle collisions in the gas transport process. The BGK schemes for
the ultra-relativistic viscous flows are also developed and two examples of
ultra-relativistic viscous flow are designed. Several 1D and 2D numerical
experiments are conducted to demonstrate that the proposed BGK schemes not only
are accurate and stable in simulating ultra-relativistic inviscid and viscous
flows, but also have higher resolution at the contact discontinuity than the
KFVS or BGK-type schemes.Comment: 41 pages, 13 figure
Numerical methods for nonlinear Dirac equation
This paper presents a review of the current state-of-the-art of numerical
methods for nonlinear Dirac (NLD) equation. Several methods are extendedly
proposed for the (1+1)-dimensional NLD equation with the scalar and vector
self-interaction and analyzed in the way of the accuracy and the time
reversibility as well as the conservation of the discrete charge, energy and
linear momentum. Those methods are the Crank-Nicolson (CN) schemes, the
linearized CN schemes, the odd-even hopscotch scheme, the leapfrog scheme, a
semi-implicit finite difference scheme, and the exponential operator splitting
(OS) schemes. The nonlinear subproblems resulted from the OS schemes are
analytically solved by fully exploiting the local conservation laws of the NLD
equation. The effectiveness of the various numerical methods, with special
focus on the error growth and the computational cost, is illustrated on two
numerical experiments, compared to two high-order accurate Runge-Kutta
discontinuous Galerkin methods. Theoretical and numerical comparisons show that
the high-order accurate OS schemes may compete well with other numerical
schemes discussed here in terms of the accuracy and the efficiency. A
fourth-order accurate OS scheme is further applied to investigating the
interaction dynamics of the NLD solitary waves under the scalar and vector
self-interaction. The results show that the interaction dynamics of two NLD
solitary waves depend on the exponent power of the self-interaction in the NLD
equation; collapse happens after collision of two equal one-humped NLD solitary
waves under the cubic vector self-interaction in contrast to no collapse
scattering for corresponding quadric case.Comment: 39 pages, 13 figure
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