arXiv subjects

Huazhong Tang

Publications and source records attributed to Huazhong Tang.

43 records · Page 3Linked to original sources

arXiv2016

Physical-constraints-preserving central discontinuous Galerkin methods for special relativistic hydrodynamics with a general equation of state

The ideal gas equation of state (EOS) with a constant adiabatic index is a poor approximation for most relativistic astrophysical flows, although it is commonly used in relativistic hydrodynamics. The paper develops high-order accurate physical-constraints-preserving (PCP) central discontinuous Galerkin (DG) methods for the one- and two-dimensional special relativistic hydrodynamic (RHD) equations with a general EOS. It is built on the theoretical analysis of the admissible states for the RHD and the PCP limiting procedure enforcing the admissibility of central DG solutions. The convexity, scaling and orthogonal invariance, and Lax-Friedrichs splitting property of the admissible state set are first proved with the aid of its equivalent form, and then the high-order central DG methods with the PCP limiting procedure and strong stability preserving time discretization are proved to preserve the positivity of the density, pressure, and specific internal energy, and the bound of the fluid velocity, maintain the high-order accuracy, and be $L^1$-stable. The accuracy, robustness, and effectiveness of the proposed methods are demonstrated by several 1D and 2D numerical examples involving large Lorentz factor, strong discontinuities, or low density or pressure etc.

math.NA↗

arXiv2016

A stochastic Galerkin method for general system of quasilinear hyperbolic conservation laws with uncertainty

This paper is concerned with generalized polynomial chaos (gPC) approximation for a general system of quasilinear hyperbolic conservation laws with uncertainty. The one-dimensional (1D) hyperbolic system is first symmetrized with the aid of left eigenvector matrix of the Jacobian matrix. Stochastic Galerkin method is then applied to derive the equations for the gPC expansion coefficients. The resulting deterministic gPC Galerkin system is proved to be symmetrically hyperbolic. This important property then allows one to use a variety of numerical schemes for spatial and temporal discretization. Here a higher-order and path-conservative finite volume WENO scheme is adopted in space, along with a third-order total variation diminishing Runge-Kutta method in time. The method is further extended to two-dimensional (2D) quasilinear hyperbolic system with uncertainty, where the symmetric hyperbolicity of the one-dimensional system is carried over via the operator splitting technique. Several 1D and 2D numerical experiments are conducted to demonstrate the accuracy and effectiveness of the proposed gPC stochastic Galerkin method.

math.NA↗

arXiv2016

A direct Eulerian GRP scheme for spherically symmetric general relativistic hydrodynamics

The paper proposes a second-order accurate direct Eulerian generalized Riemann problem (GRP) scheme for the spherically symmetric general relativistic hydrodynamical (RHD) equations and a second-order accurate discretization for the spherically symmetric Einstein (SSE) equations. The former is directly using the Riemann invariants and the Runkine-Hugoniot jump conditions to analytically resolve the left and right nonlinear waves of the local GRP in the Eulerian formulation together with the local change of the metrics to obtain the limiting values of the time derivatives of the conservative variables along the cell interface and the numerical flux for the GRP scheme. While the latter utilizes the energy-momentum tensor obtained in the GRP solver to evaluate the fluid variables in the SSE equations and keeps the continuity of the metrics at the cell interfaces. Several numerical experiments show that the GRP scheme can achieve second-order accuracy and high resolution, and is effective for spherically symmetric general RHD problems.

math.NA↗

arXiv2015

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.

math.NA↗

arXiv2013

Multi-hump solitary waves of nonlinear Dirac equation

This paper concentrates on a (1+1)-dimensional nonlinear Dirac (NLD) equation with a general self-interaction, being a linear combination of the scalar, pseudoscalar, vector and axial vector self-interactions to the power of the integer $k+1$. The solitary wave solutions to the NLD equation are analytically derived, and the upper bounds of the hump number in the charge, energy and momentum densities for the solitary waves are proved in theory. The results show that: (1) for a given integer $k$, the hump number in the charge density is not bigger than $4$, while that in the energy density is not bigger than $3$; (2) those upper bounds can only be achieved in the situation of higher nonlinearity, namely, $k\in\{5,6,7,\cdots \}$ for the charge density and $k\in\{3,5,7,\cdots\}$ for the energy density; (3) the momentum density has the same multi-hump structure as the energy density; (4) more than two humps (resp. one hump) in the charge (resp. energy) density can only happen under the linear combination of the pseudoscalar self-interaction and at least one of the scalar and vector (or axial vector) self-interactions. Our results on the multi-hump structure will be interesting in the interaction dynamics for the NLD solitary waves.

nlin.SI↗

arXiv2013

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.

physics.comp-ph↗

arXiv2006

Interaction for Solitary Waves with a Phase Difference in a Nonlinear Dirac Model

This paper presents a further numerical study of the interaction dynamics for solitary waves in a nonlinear Dirac field with scalar self-interaction by using a fourth order accurate Runge-Kutta discontinuous Galerkin method. Our experiments are conducted on the Dirac solitary waves with a phase difference. Some interesting phenomena are observed: (a) full repulsion in binary and ternary collisions and its dependence on the distance between initial waves; (b) repulsing first, attracting afterwards, and then collapse in binary and ternary collisions of initially resting two-humped waves; (c) one-overlap interaction and two-overlap interaction in ternary collisions of initially resting waves.

nlin.SI↗