Dimensional Conservation Laws Research Articles

A Third Order Adaptive ADER Scheme for One Dimensional Conservation Laws

AbstractWe introduce a third order adaptive mesh method to arbitrary high order Godunov approach. Our adaptive mesh method consists of two parts, i.e., mesh-redistribution algorithm and solution algorithm. The mesh-redistribution algorithm is derived based on variational approach, while a new solution algorithm is developed to preserve high order numerical accuracy well. The feature of proposed Adaptive ADER scheme includes that 1). all simulations in this paper are stable for large CFL number, 2). third order convergence of the numerical solutions is successfully observed with adaptive mesh method, and 3). high resolution and non-oscillatory numerical solutions are obtained successfully when there are shocks in the solution. A variety of numerical examples show the feature well.

Totally Volume Integral of Fluxes for Discontinuous Galerkin Method (TVI-DG) I-Unsteady Scalar One Dimensional Conservation Laws

The volume integral of Riemann flux in the discontinuous Galerkin (DG) method is introduced in this paper. The boundaries integrals of the fluxes (Riemann flux) are transformed into volume integral. The new family of DG method is accomplished by applying divergence theorem to the boundaries integrals of the flux. Therefore, the (DG) method is independent of the boundaries integrals of fluxes (Riemann flux) at the cell (element) boundaries as in classical (DG) methods. The modified streamline upwind Petrov-Galerkin method is used to capture the oscillation of unphysical flow for shocked flow problems. The numerical results of applying totally volume integral discontinuous Galerkin method (TVI-DG) are presented to unsteady scalar hyperbolic equations (linear convection equation, inviscid Burger's equation and Buckley-Leverett equation) for one dimensional case. The numerical finding of this scheme is very accurate as compared with other high order schemes as the weighted compact finite difference method WCOMP.

A two-dimensional glimm type scheme on cauchy problem of two-dimensional scalar conservation law

In this paper, we construct a new two-dimensional convergent scheme to solve Cauchy problem of following two-dimensional scalar conservation law {∂tu+∂xf(u)+∂yg(u)=0,u(x,y,0)=u0(x,y). In which initial data can be unbounded. Although the existence and uniqueness of the weak entropy solution are obtained, little is known about how to investigate two-dimensional or higher dimensional conservation law by the schemes based on wave interaction of 2D Riemann solutions and their estimation. So we construct such scheme in our paper and get some new results.

Arbitrary Lagrangian-Eulerian discontinuous Galerkin method for conservation laws: Analysis and application in one dimension

In this paper, we develop and analyze an arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) method with a time-dependent approximation space for one dimensional conservation laws, which satisfies the geometric conservation law. For the semi-discrete ALE-DG method, when applied to nonlinear scalar conservation laws, a cell entropy inequality, L2 stability and error estimates are proven. More precisely, we prove the sub-optimal (k + 12) convergence for monotone fluxes, and optimal (k + 1) convergence for an upwind flux, when a piecewise P k polynomial approximation space is used. For the fully-discrete ALE-DG method, the geometric conservation law and the local maximum principle are proven. Moreover we state conditions for slope limiters, which ensure total variation stability of the method. Numerical examples show the capability of the method.

Nonlinear wave propagation in cylindrical air-filled tubes

Three dimensional conservation laws are reformulated in a dimensionless form for a cylindrical air-filled tube. Three nondimensional numbers therefore appear, and are used to evaluate the order of magnitude of each term. Simplifying leads to two quasi-unidimensional generalized Burgers equations, which take into account both nonlinear phenomena and viscothermal losses. Special attention is paid to the validity of the different approximations we proposed. The method of multiple scales indicates that the interaction between the two opposite propagative waves is noncumulative and remains a local effect. The two Burgers equations are therefore globally independent. A numerical computation in the frequency domain solves the Burgers equations in the case of a stationary wave. The signal is supposed to be known at the two ends of the tube (or impedance instead of signal for one extremity). The computation uses the harmonic balance method to calculate the two opposite propagative waves which makes it possible to verify the right conditions in the two extremities. The signal everywhere in the tube is so pieced together. Some experiments are performed with a siren as source in one extremity, and measurements are compared with theoretical results.

A Functional Integral Approach to Shock Wave Solutions of the Euler Equations with Spherical Symmetry (II)

This is a continuation of [15]. As is well known, one dimensional conservation laws without source term have been extensively investigated after the foundamental paper of J. Glimm [3]. And those with integrable source terms were solved by Liu [6, 7], etc. For higher dimensional case with spherical symmetry, Makino [8] first proved a linear growth rate for solutions whenP(ρ)=σ2ργ, whereγ=1. But in order to get global existence forγ≠1 and the decay property, we need to find a uniform bound for the approximate solutions. In [15], we introduced a new norm and a functional integral approach to prove a uniform bound for a model problem of Euler equation inR3with spherical symmetry. In order to overcome the geometric effects of spherical symmetry which leads to a non-integrable source term, we considered an infinite reflection problem and solved it by considering the cancellations between reflections of different orders. In this paper, we consider a system which describes the isentropic and spherically symmetric motion of gas flow surrounding a solid star with radius 1 and massM. It is interesting to note that the wave curves for this problem are no longer continuous and there is an extra term in the wave interaction estimates. By introducing a new norm, we prove a similar result as [15]. In the Appendix, we present a local existence theorem forγ≠1 which was also obtained by Makino [9] by different method. And we extend the results to the cases with different boundary conditions.

SOME FUNDAMENTAL CONCEPTS ABOUT SYSTEM OF TWO SPATIAL DIMENSIONAL CONSERVATION LAWS

Lax, Gel'fand, Oleinik and others have established some fundamental concepts about system of one spatial dimensional conservation laws, such as hyperbolicity, genuine nonlinearity (convexity), linear degenerate, Rankine–Hugoniot condition, stability condition (entropy condition), stability condition (E) and so on. We extend these concepts to system of two spatial dimension case in this note. It is a necessary preliminary for advance study.