Dimension by Dimension Finite Volume HWENO Method for Hyperbolic Conservation Laws

In this paper,we propose a finite volume Hermite weighted essentially non-oscillatory(HWENO)method based on the dimension by dimension framework to solve hyperbolic conservation laws.It can maintain the high accuracy in the smooth region and obtain the high resolution solution when the discontinuity appears,and it is compact which will be good for giving the numerical boundary conditions.Furthermore,it avoids complicated least square procedure when we implement the genuine two dimensional(2D)finite volume HWENO reconstruction,and it can be regarded as a generalization of the one dimensional(1D)HWENO method.Extensive numerical tests are performed to verify the high resolution and high accuracy of the scheme.

Nonuniform Dependence on the Initial Data for Solutions of Conservation Laws

In this paper,we study systems of conservation laws in one space dimension.We prove that for classical solutions in Sobolev spaces H^(s),with s>3/2,the data-to-solution map is not uniformly continuous.Our results apply to all nonlinear scalar conservation laws and to nonlinear hyperbolic systems of two equations.

Darboux transformation,infinite conservation laws,and exact solutions for the nonlocal Hirota equation with variable coefficients

This article presents the construction of a nonlocal Hirota equation with variable coefficients and its Darboux transformation.Using zero-seed solutions,1-soliton and 2-soliton solutions of the equation are constructed through the Darboux transformation,along with the expression for N-soliton solutions.Influence of coefficients that are taken as a function of time instead of a constant,i.e.,coefficient functionδ(t),on the solutions is investigated by choosing the coefficient functionδ(t),and the dynamics of the solutions are analyzed.This article utilizes the Lax pair to construct infinite conservation laws and extends it to nonlocal equations.The study of infinite conservation laws for nonlocal equations holds significant implications for the integrability of nonlocal equations.

Regularity of Fluxes in Nonlinear Hyperbolic Balance Laws

This paper addresses the issue of the formulation of weak solutions to systems of nonlinear hyperbolic conservation laws as integral balance laws.The basic idea is that the“meaningful objects”are the fluxes,evaluated across domain boundaries over time intervals.The fundamental result in this treatment is the regularity of the flux trace in the multi-dimensional setting.It implies that a weak solution indeed satisfies the balance law.In fact,it is shown that the flux is Lipschitz continuous with respect to suitable perturbations of the boundary.It should be emphasized that the weak solutions considered here need not be entropy solutions.Furthermore,the assumption imposed on the flux f(u)is quite minimal-just that it is locally bounded.

On High-Resolution Entropy-Consistent Flux with Slope Limiter for Hyperbolic Conservation Laws

This paper proposes a new version of the high-resolution entropy-consistent(EC-Limited)flux for hyperbolic conservation laws based on a new minmod-type slope limiter.Firstly,we identify the numerical entropy production,a third-order differential term deduced from the previous work of Ismail and Roe[11].The corresponding dissipation term is added to the original Roe flux to achieve entropy consistency.The new,resultant entropy-consistent(EC)flux has a general and explicit analytical form without any corrective factor,making it easy to compute and a less-expensive method.The inequality constraints are imposed on the standard piece-wise quadratic reconstruction to enforce the pointwise values of bounded-type numerical solutions.We design the new minmod slope limiter as combining two separate limiters for left and right states.We propose the EC-Limited flux by adding this reconstruction data method to the primitive variables rather than to the conservative variables of the EC flux to preserve the equilibrium of the primitive variables.These resulting fluxes are easily applied to general hyperbolic conservation laws while having attractive features:entropy-stable,robust,and non-oscillatory.To illustrate the potential of these proposed fluxes,we show the applications to the Burgers equation and the Euler equations.

Existence of Entropy Solution for Degenerate Parabolic-Hyperbolic Problem Involving p(x)-Laplacian with Neumann Boundary Condition

We consider a strongly non-linear degenerate parabolic-hyperbolic problem with p(x)-Laplacian diffusion flux function. We propose an entropy formulation and prove the existence of an entropy solution.

New High-Order Numerical Methods for Hyperbolic Systems of Nonlinear PDEs with Uncertainties

In this paper,we develop new high-order numerical methods for hyperbolic systems of nonlinear partial differential equations(PDEs)with uncertainties.The new approach is realized in the semi-discrete finite-volume framework and is based on fifth-order weighted essentially non-oscillatory(WENO)interpolations in(multidimensional)random space combined with second-order piecewise linear reconstruction in physical space.Compared with spectral approximations in the random space,the presented methods are essentially non-oscillatory as they do not suffer from the Gibbs phenomenon while still achieving high-order accuracy.The new methods are tested on a number of numerical examples for both the Euler equations of gas dynamics and the Saint-Venant system of shallow-water equations.In the latter case,the methods are also proven to be well-balanced and positivity-preserving.

Tensile Shock Physics in Compressible Thermoviscoelastic Solid Medium

This paper addresses tensile shock physics in thermoviscoelastic (TVE) solids without memory. The mathematical model is derived using conservation and balance laws (CBL) of classical continuum mechanics (CCM), incorporating the contravariant second Piola-Kirchhoff stress tensor, the covariant Green’s strain tensor, and its rates up to order n. This mathematical model permits the study of finite deformation and finite strain compressible deformation physics with an ordered rate dissipation mechanism. Constitutive theories are derived using conjugate pairs in entropy inequality and the representation theorem. The resulting mathematical model is both thermodynamically and mathematically consistent and has closure. The solution of the initial value problems (IVPs) describing evolutions is obtained using a variationally consistent space-time coupled finite element method, derived using space-time residual functional in which the local approximations are in hpk higher-order scalar product spaces. This permits accurate description problem physics over the discretization and also permits precise a posteriori computation of the space-time residual functional, an accurate measure of the accuracy of the computed solution. Model problem studies are presented to demonstrate tensile shock formation, propagation, reflection, and interaction. A unique feature of this research is that tensile shocks can only exist in solid matter, as their existence requires a medium to be elastic (presence of strain), which is only possible in a solid medium. In tensile shock physics, a decrease in the density of the medium caused by tensile waves leads to shock formation ahead of the wave. In contrast, in compressive shocks, an increase in density and the corresponding compressive waves result in the formation of compression shocks behind of the wave. Although these are two similar phenomena, they are inherently different in nature. To our knowledge, this work has not been reported in the published literature.

Residual symmetry, interaction solutions, and conservation laws of the(2+1)-dimensional dispersive long-wave system

We explore the （2＋l）-dimensional dispersive long-wave （DLW） system. From the standard truncated Painleve expansion, the Baicklund transformation （BT） and residual symmetries of this system are derived. The introduction to an appropriate auxiliary dependent variable successfully localizes the residual symmetries to Lie point symmetries. In particular, it is verified that the （2＋l）-dimensional DLW system is consistent Riccati expansion （CRE） solvable. If the special form of （CRE）-consistent tanh-function expansion （CTE） is taken, the soliton-cnoidal wave solutions and corresponding images can be explicitly given. Furthermore, the conservation laws of the DLW system are investigated with symmetries and Ibragimov theorem.

New Exact Solutions and Conservation Laws to （3＋1）-Dimensional Potential-YTSF Equation

Using the modified find some new exact solutions to Lie point symmetry groups and also get conservation laws, of the CK＇s direct method, we build the relationship between new solutions and old ones and the （3＋1）-dimensional potentiaial-YTSF equation. Baaed on the invariant group theory, Lie symmetries of the （3＋1）-dimensional potential-YTSF equation are obtained. We equation with the given Lie symmetry.

A local pseudo arc-length method for hyperbolic conservation laws

A local pseudo arc-length method（LPALM）for solving hyperbolic conservation laws is presented in this paper.The key idea of this method comes from the original arc-length method,through which the critical points are bypassed by transforming the computational space.The method is based on local changes of physical variables to choose the discontinuous stencil and introduce the pseudo arc-length parameter,and then transform the governing equations from physical space to arc-length space.In order to solve these equations in arc-length coordinate,it is necessary to combine the velocity of mesh points in the moving mesh method,and then convert the physical variable in arclength space back to physical space.Numerical examples have proved the effectiveness and generality of the new approach for linear equation,nonlinear equation and system of equations with discontinuous initial values.Non-oscillation solution can be obtained by adjusting the parameter and the mesh refinement number for problems containing both shock and rarefaction waves.

Symmetry Reductions, Exact Solutions and Conservation Laws of Asymmetric Nizhnik-Novikov-Veselov Equation

By applying a direct symmetry method, we get the symmetry of the asymmetric Nizhnik-Novikov-Veselov equation （ANNV）. Taking the special case, we have a finite-dimensional symmetry. By using the equivalent vector of the symmetry, we construct an eight-dimensional symmetry algebra and get the optimal system of group-invariant solutions. To every case of the optimal system, we reduce the ANNV equation and obtain some solutions to the reduced equations. Furthermore, we find some new explicit solutions of the ANNV equation. At last, we give the conservation laws of the ANNV equation.

Integrating Factors and Conservation Laws of Generalized Birkhoff System Dynamics in Event Space

In this paper, the conservation laws of generalized Birkhoff system in event space are studied by using the method of integrating factors. Firstly, the generalized Pfaff-Birkhoff principle and the generalized Birkhoff equations are established, and the definition of the integrating factors for the system is given. Secondly, based on the concept of integrating factors, the conservation theorems and their inverse for the generalized Birkhoff system in the event space are presented in detail, and the relation between the conservation laws and the integrating factors of the system is obtained and the generalized Killing equations for the determination of the integrating factors are given. Finally, an example is given to illustrate the application of the results.

Integrating Factors and Conservation Laws for Relativistic Mechanical System

In this paper, we present a new method to construct the conservation laws for relativistic mechanical systems by finding corresponding integrating factors. First, the Lagrange equations of relativistic mechanical systems are established, and the definition of integrating factors of the systems is given; second, the necessary conditions for the existence of conserved quantities of the relativistic mechanical systems are studied in detail, and the relation between the conservation laws and the integrating factors of the systems is obtained and the generalized Killing equations for the determination of the integrating factors are given; finally, the conservation theorem and its inverse for the systems are established, and an example is given to illustrate the application of the results.

NEW STRUCTURES FOR NON-SELFSIMILAR SOLUTIONS OF MULTI-DIMENSIONAL CONSERVATION LAWS

In this article, we get non-selfsimilar elementary waves of the conservation laws in another kind of view, which is different from the usual self-similar transformation. The solution has different global structure. This article is divided into three parts. The first part is introduction. In the second part, we discuss non-selfsimilar elementary waves and their interactions of a class of twodimensional conservation laws. In this case, we consider the case that the initial discontinuity is parabola with u＋ 〉 0, while explicit non-selfsirnilar rarefaction wave can be obtained. In the second part, we consider the solution structure of case u＋ 〈 0. The new solution structures are obtained by the interactions between different elementary waves, and will continue to interact with other states. Global solutions would be very different from the situation of one dimension.

Conservation laws of the generalized nonlocal nonlinear Schrodinger equation

The derivations of several conservation laws of the generalized nonlocal nonlinear Schrodinger equation are presented. These invaxiants are the number of particles, the momentum, the angular momentum and the Hamiltonian in the quantum mechanical analogy. The Lagrangian is also presented.

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (X) -MASTER BALANCE LAW

Through a comparison between the expressions of master balance laws and the conservation laws derived by Noether＇s theorem, a unified master balance law and six physically possible balance equations for micropolar continuum mechanics are naturally deduced. Among them, by extending the well-known conventional concept of energymomentum tensor, the rather general conservation laws and balance equations named after energy-momentum, energy-angular momentum and energy-energy are obtained. It is clear that the forms of the physical field quantities in the master balance law for the last three cases could not be assumed directly by perceiving through the intuition. Finally, some existing results are reduced immediately as special cases.

Super spectral viscosity method for nonlinear conservation laws

In this paper, the super spectral viscosity （SSV） method is developed by introducing a spectrally small amount of high order regularization which is only activated on high frequencies. The resulting SSV approximation is stable and convergent to the exact entropy solution. A Gegenbauer-Chebyshev post-processing for the SSV solution is proposed to remove the spurious oscillations at the disconti-nuities and recover accuracy from the spectral approximation. The ssv method is applied to the scahr periodic Burgers equation and the one-dimensional system of Euler equations of gas dynamics. The numerical results exhibit high accuracy and resolution to the exact entropy solution,

POINTWISE CONVERGENCE RATE OF VANISHING VISCOSITY APPROXIMATIONS FOR SCALAR CONSERVATION LAWS WITH BOUNDARY

This article is concerned with the pointwise error estimates for vanishing vis- cosity approximations to scalar convex conservation laws with boundary.By the weighted error function and a bootstrap extrapolation technique introduced by Tadmor-Tang,an optimal pointwise convergence rate is derived for the vanishing viscosity approximations to the initial-boundary value problem for scalar convex conservation laws,whose weak entropy solution is piecewise C 2 -smooth with interaction of elementary waves and the ...

Conservation Laws and Soliton Solutions for Generalized Seventh Order KdV Equation

With the assistance of the symbolic computation system Maple,rich higher order polynomial-type conservation laws and a sixth order t/x-dependent conservation law are constructed for a generalized seventh order nonlinear evolution equation by using a direct algebraic method.From the compatibility conditions that guaranteeing the existence of conserved densities,an integrable unnamed seventh order KdV-type equation is found.By introducing some nonlinear transformations,the one-,two-,and three-solition solutions as well as the solitary wave solutions are obtained.