THE OPTIMAL LARGE TIME BEHAVIOR OF3D QUASILINEAR HYPERBOLIC EQUATIONS WITH NONLINEAR DAMPING

We are concerned with the large-time behavior of 3D quasilinear hyperbolic equations with nonlinear damping.The main novelty of this paper is two-fold.First,we prove the optimal decay rates of the second and third order spatial derivatives of the solution,which are the same as those of the heat equation,and in particular,are faster than ones of previous related works.Second,for well-chosen initial data,we also show that the lower optimal L^(2) convergence rate of the k(∈[0,3])-order spatial derivatives of the solution is(1+t)^(-(2+2k)/4).Therefore,our decay rates are optimal in this sense.The proofs are based on the Fourier splitting method,low-frequency and high-frequency decomposition,and delicate energy estimates.

Oscillation Theorem of Systems of Quasilinear Impulsive Delay Hyperbolic Equations

In this paper, oscillatory properties for solutions of the systems of certain quasilinear impulsive delay hyperbolic equations with nonlinear diffusion coefficient are investigated. A sufficient criterion for oscillations of such systems is obtained.

A REDUCED FE FORMULATION BASED ON POD METHOD FOR HYPERBOLIC EQUATIONS

A proper orthogonal decomposition（POD） method was successfully used in the reduced-order modeling of complex systems.In this paper,we extend the applications of POD method,namely,apply POD method to a classical finite element（FE） formulation for second-order hyperbolic equations with real practical applied background,establish a reduced FE formulation with lower dimensions and high enough accuracy,and provide the error estimates between the reduced FE solutions and the classical FE solutions and the implementation of algorithm for solving reduced FE formulation so as to provide scientific theoretic basis for service applications.Some numerical examples illustrate the fact that the results of numerical computation are consistent with theoretical conclusions.Moreover,it is shown that the reduced FE formulation based on POD method is feasible and efficient for solving FE formulation for second-order hyperbolic equations.

A POD REDUCED-ORDER SPDMFE EXTRAPOLATING ALGORITHM FOR HYPERBOLIC EQUATIONS

In this article, a proper orthogonal decomposition （POD） method is used to study a classical splitting positive definite mixed finite element （SPDMFE） formulation for second- order hyperbolic equations. A POD reduced-order SPDMFE extrapolating algorithm with lower dimensions and sufficiently high accuracy is established for second-order hyperbolic equations. The error estimates between the classical SPDMFE solutions and the reduced-order SPDMFE solutions obtained from the POD reduced-order SPDMFE extrapolating algorithm are provided. The implementation for solving the POD reduced-order SPDMFE extrapolating algorithm is given. Some numerical experiments are presented illustrating that the results of numerical computation are consistent with theoretical conclusions, thus validating that the POD reduced-order SPDMFE extrapolating algorithm is feasible and efficient for solving second-order hyperbolic equations.

OSCILLATION CRITERIA OF NEUTRAL TYPE IMPULSIVE HYPERBOLIC EQUATIONS

In this paper, oscillatory properties of all solutions for neutral type impulsive hyperbolic equations with several delays under the Robin boundary condition are investigated and several new sufficient conditions for oscillation are presented.

ADER Methods for Hyperbolic Equations with a Time-Reconstruction Solver for the Generalized Riemann Problem: the Scalar Case

The ADER approach to solve hyperbolic equations to very high order of accuracy has seen explosive developments in the last few years,including both methodological aspects as well as very ambitious applications.In spite of methodological progress,the issues of efficiency and ease of implementation of the solution of the associated generalized Riemann problem(GRP)remain the centre of attention in the ADER approach.In the original formulation of ADER schemes,the proposed solution procedure for the GRP was based on(i)Taylor series expansion of the solution in time right at the element interface,(ii)subsequent application of the Cauchy-Kowalewskaya procedure to convert time derivatives to functionals of space derivatives,and(iii)solution of classical Riemann problems for high-order spatial derivatives to complete the Taylor series expansion.For realistic problems the Cauchy-Kowalewskaya procedure requires the use of symbolic manipulators and being rather cumbersome its replacement or simplification is highly desirable.In this paper we propose a new class of solvers for the GRP that avoid the Cauchy-Kowalewskaya procedure and result in simpler ADER schemes.This is achieved by exploiting the history of the numerical solution that makes it possible to devise a time-reconstruction procedure at the element interface.Still relying on a time Taylor series expansion of the solution at the interface,the time derivatives are then easily calculated from the time-reconstruction polynomial.The resulting schemes are called ADER-TR.A thorough study of the linear stability properties of the linear version of the schemes is carried out using the von Neumann method,thus deducing linear stability regions.Also,via careful numerical experiments,we deduce stability regions for the corresponding non-linear schemes.Numerical examples using the present simplified schemes of fifth and seventh order of accuracy in space and time show that these compare favourably with conventional ADER methods.This paper is restricted to the one-dimensional scalar case with source term,but preliminary results for the one-dimensional Euler equations indicate that the time-reconstruction approach offers significant advantages not only in terms of ease of implementation but also in terms of efficiency for the high-order range schemes.

Oscillatory Criteria for a Class of Boundary Value Problem of Nonlinear Hyperbolic Equations *L

Aim To study a class of boundary value problem of hyperbolic partial functional differential equations with continuous deviating arguments. Methods An averaging technique was used. The multi dimensional problem was reduced to a one dimensional oscillation problem for ordinary differential equations or inequalities. Results and Conclusion The known results of oscillation of solutions for a class of boundary value problem of hyperbolic partial functional differential equations with discrete deviating arguments are generalized, and the oscillatory criteria of solutions for such equation with two kinds of boundary value conditions are obtained.

Existence and asymptotic behavior for systems of nonlinear hyperbolic equations

The initial-boundary value problem for a class of nonlinear hyperbolic equations system in bounded domain is studied. The existence of global solutions for this problem is proved by constructing a stable set, and obtain the asymptotic stability of global solutions by means of a difference inequality.

AN ADI GALERKIN METHOD WITH MOVING FINITE ELEMENT SPACES FOR A CLASS OF SECOND-ORDER HYPERBOLIC EQUATIONS

An alternating direction implicit (ADI) Galerkin method with moving finite element spaces is formulated for a class of second order hyperbolic equations in two space variables. A priori H 1 error estimate is derived.

AENO:a Novel Reconstruction Method in Conjunction with ADER Schemes for Hyperbolic Equations

In this paper,we present a novel spatial reconstruction scheme,called AENO,that results from a special averaging of the ENO polynomial and its closest neighbour,while retaining the stencil direction decided by the ENO choice.A variant of the scheme,called m-AENO,results from averaging the modified ENO(m-ENO)polynomial and its closest neighbour.The concept is thoroughly assessed for the one-dimensional linear advection equation and for a one-dimensional non-linear hyperbolic system,in conjunction with the fully discrete,high-order ADER approach implemented up to fifth order of accuracy in both space and time.The results,as compared to the conventional ENO,m-ENO and WENO schemes,are very encouraging.Surprisingly,our results show that the L_(1)-errors of the novel AENO approach are the smallest for most cases considered.Crucially,for a chosen error size,AENO turns out to be the most efficient method of all five methods tested.

Further Oscillation Results for a Class of Hyperbolic Equations with Continuous Distributed Deviating Arguments

A class of hyperbolic equations with continuous distributed deviating arguments is considered and its oscillation theorems are discussed.These theorems are of higher degree of generality and deal with the cases which are not covered by the known criteria.Particularly,these criteria extend and unify a number of existing results.

Negative Norm Estimates for Arbitrary Lagrangian-Eulerian Discontinuous Galerkin Method for Nonlinear Hyperbolic Equations

In this paper,we present the negative norm estimates for the arbitrary Lagrangian-Eulerian discontinuous Galerkin(ALE-DG)method solving nonlinear hyperbolic equations with smooth solutions.The smoothness-increasing accuracy-conserving(SIAC)filter is a post-processing technique to enhance the accuracy of the discontinuous Galerkin(DG)solutions.This work is the essential step to extend the SIAC filter to the moving mesh for nonlinear problems.By the post-processing theory,the negative norm estimates are vital to get the superconvergence error estimates of the solutions after post-processing in the L2 norm.Although the SIAC filter has been extended to nonuniform mesh,the analysis of fil-tered solutions on the nonuniform mesh is complicated.We prove superconvergence error estimates in the negative norm for the ALE-DG method on moving meshes.The main dif-ficulties of the analysis are the terms in the ALE-DG scheme brought by the grid velocity field,and the time-dependent function space.The mapping from time-dependent cells to reference cells is very crucial in the proof.The numerical results also confirm the theoreti-cal proof.

A Class of the Quasilinear Hyperbolic Equations with the Inhomogenous Terms

In this paper,we discuss a class of the quasillinear hyperbolic equations with the inhomogeneous terms: u_■+σ(v)+2α(t)u=0.v_■-u-0 Under the certain of hypothesis.we prove the globally existence theorems of the smooth solutions for its Cauchy problem.

Potential symmetries and conservation laws for generalized quasilinear hyperbolic equations

Based on the Lie group method, the potential symmetries and invariant solutions for generalized quasilinear hyperbolic equations are studied. To obtain the invariant solutions in an explicit form, the physically interesting situations with potential symmetries are focused on, and the conservation laws for these equations in three physi- cally interesting cases are found by using the partial Lagrangian approach.

Superconvergence analysis of the finite element method for nonlinear hyperbolic equations with nonlinear boundary condition

This paper discusses the semidiscrete finite element method for nonlinear hyperbolic equations with nonlinear boundary condition. The superclose property is derived through interpolation instead of the nonlinear H^1 projection and integral identity technique. Meanwhile, the global superconvergence is obtained based on the interpolated postprocessing techniques.

Numerical Analysis of Upwind Difference Schemes for Two-Dimensional First-Order Hyperbolic Equations with Variable Coefficients

In this paper, we consider the initial-boundary value problem of two-dimensional first-order linear hyperbolic equation with variable coefficients. By using the upwind difference method to discretize the spatial derivative term and the forward and backward Euler method to discretize the time derivative term, the explicit and implicit upwind difference schemes are obtained respectively. It is proved that the explicit upwind scheme is conditionally stable and the implicit upwind scheme is unconditionally stable. Then the convergence of the schemes is derived. Numerical examples verify the results of theoretical analysis.

Reduced-order proper orthogonal decomposition extrapolating finite volume element format for two-dimensional hyperbolic equations

This paper is concerned with establishing a reduced-order extrapolating fi- nite volume element （FVE） format based on proper orthogonal decomposition （POD） for two-dimensional （2D） hyperbolic equations. For this purpose, a semi discrete variational format relative time and a fully discrete FVE format for the 2D hyperbolic equations are built, and a set of snapshots from the very few FVE solutions are extracted on the first very short time interval. Then, the POD basis from the snapshots is formulated, and the reduced-order POD extrapolating FVE format containing very few degrees of freedom but holding sufficiently high accuracy is built. Next, the error estimates of the reduced-order solutions and the algorithm procedure for solving the reduced-order for- mat are furnished. Finally, a numerical example is shown to confirm the correctness of theoretical conclusions. This means that the format is efficient and feasible to solve the 2D hyperbolic equations.

Oscillation Criteria for Impulsive Hyperbolic Equations of Neutral Type

Oscillation properties of the solutions of impulsive hyperbolic equations are investigated by the method of differential inequalities.

A POSTERIORI ERROR ESTIMATE OF THE DSD METHOD FOR FIRST-ORDER HYPERBOLIC EQUATIONS

A posteriori error estimate of the discontinuous-streamline diffusion method for first-order hyperbolic equations was presented, which can be used to adjust space mesh reasonably. A numerical example is given to illustrate the accuracy and feasibility of this method.

Iterative method of solving Cauchy problem with reproducing kernel for nonlinear hyperbolic equations

Presents the iterative method of solving Cauchy problem with reproducing kernel for nonlinear hyperbolic equations, and the application of the computational technique of reproducing kernel space to simplify, the iterative computation and increase the convergence rate and points out that this method is still effective. Even if the initial condition is discrete.