Quasilinear Hyperbolic Problems Research Articles

Approximation of Quasilinear Hyperbolic Problems with Discontinuous Coefficients: An Optimal Error Estimate

We develop a theoretical framework for the approximation of a class of second-order quasilinear hyperbolic interface problems on quadratic finite element. Time discretization based on linearized implicit difference scheme with degree two polynomials for interface approximation is proposed. Sufficient conditions, on the input data, that guarantee the existence of a unique solution are given. Under these assumptions, the stability of the scheme is established and convergence rate of optimal order is proved. It is assumed that the interface is arbitrary but smooth.

Error analysis of implicit Euler methods for quasilinear hyperbolic evolution equations

In this paper we study the convergence of the semi-implicit and the implicit Euler methods for the time integration of abstract, quasilinear hyperbolic evolution equations. The analytical framework considered here includes certain quasilinear Maxwell's and wave equations as special cases. Our analysis shows that the Euler approximations are well-posed and convergent of order one. The techniques will be the basis for the future investigation of higher order time integration methods and full discretizations of certain quasilinear hyperbolic problems.

Smooth solutions of the one-dimensional compressible Euler equation with gravity

We study one-dimensional motions of polytropic gas governed by the compressible Euler equations. The problem on the half space under a constant gravity gives an equilibrium which has free boundary touching the vacuum and the linearized approximation at this equilibrium gives time periodic solutions. But it is difficult to justify the existence of long-time true solutions for which this time periodic solution is the first approximation. The situation is in contrast to the problem of free motions without gravity. The reason is that the usual iteration method for quasilinear hyperbolic problem cannot be used because of the loss of regularities which causes from the touch with the vacuum. Due to this reason, we try to find a family of solutions expanded by a small parameter and apply the Nash–Moser Theorem to justify this expansion. Note that the application of Nash–Moser Theorem is necessary for the sake of conquest of the trouble with loss of regularities, and the justification of the applicability requires a very delicate analysis of the problem.

P.D.E.s with hysteresis 30 years later

Continuous and discontinuous hysteresis operators are first reviewed in general. The Duhem model, the generalized play, the (delayed) relay and the Preisach model are outlined, as well as vector extensions of the two latter models. &nbsp Two examples of initial- and boundary-value problems for P.D.E.s with hysteresis are then illustrated. Well-posedness is proved for quasilinear parabolic problems with either continuous or discontinuous hysteresis. Existence of a weak solution is shown for second-order quasilinear hyperbolic problems with discontinuous hysteresis.

Deterministic homogenization of quasilinear damped hyperbolic equations

Deterministic homogenization is studied for quasilinear monotone hyperbolic problems with a linear damping term. It is shown by the sigma-convergence method that the sequence of solutions to a class of multi-scale highly oscillatory hyperbolic problems converges to the solution to a homogenized quasilinear hyperbolic problem.

Characteristic boundary layers for parabolic perturbations of quasilinear hyperbolic problems

In this paper, we study the existence and nonlinear stability of the totally characteristic boundary layer for the quasilinear equations with positive definite viscosity matrix under the assumption that the boundary matrix vanishes identically on the boundary x = 0 . We carry out a series of weighted estimates to the boundary layer equations—Prandtl type equations to get the regularity and the far field behavior of the solutions. This allows us to perform a weighted energy estimate for the error equation to prove the stability of the boundary layers. The stability result finally implies the asymptotic limit of the viscous solutions.

G-convergence and homogenization of monotone damped hyperbolic equations

Multiscale stochastic homogenization is studied for quasilinear hyperbolic problems. We consider the asymptotic behaviour of a sequence of realizations of the form ${\frac{\partial^2 u^\omega_{\varepsilon}}{\partial t^2}} - \mathrm{div}\left(a\left(T_1(\frac{x}{\varepsilon_1})\omega_1, T_2(\frac{x}{\varepsilon_2})\omega_2 ,t, D u^\omega_{\varepsilon}\right)\right)-\Delta(\frac{\partial u^\omega_{\varepsilon}}{\partial t}) +G\left(T_3(\frac{x}{\varepsilon_3})\omega_3 ,t,\frac{\partial u^\omega_{\varepsilon}}{\partial t}\right)=f$. It is shown, under certain structure assumptions on the random maps $a\left(\omega_1,\omega_2,t,\xi\right)$ and $G\left(\omega_3,t,\eta\right)$, that the sequence $\{u^\omega_\epsilon\}$ of solutions converges weakly in $L^p(0,T;W^{1,p}_0(\Omega))$ to the solution $u$ of the homogenized problem ${\frac{\partial^2 u}{\partial t^2}} - \mathrm{div}\left( b \left( t,D u \right)\right)-\Delta(\frac{\partial u}{\partial t})+{\overline G}(t,\frac{\partial u}{\partial t}) = f$.

CONVERGENCE OF QUASI-LINEAR HYPERBOLIC EQUATIONS

Multiscale stochastic homogenization is studied for quasilinear monotone hyperbolic problems with a linear damping term. It is shown by classical G-convergence methods that the sequence of solutions to a class of multi-scale highly oscillatory (possibly random) hyperbolic problems converges in the appropriate Sobolev space to the solution to a homogenized quasilinear hyperbolic problem.

Galerkin alternating-direction method for a kind of second-order quasi-linear hyperbolic problems

A kind of second-order quasi-linear hyperbolic equation is firstly transformed into a first-order system of equations, then the Galerkin alternating-direction procedure for the system is derived. The optimal order estimates in H 1 norm and L 2 norm of the procedure are obtained respectively by using the theory and techniques of priori estimate of differential equations. The numerical experiment is also given to support the theoretical analysis. Comparing the results of numerical example with the theoretical analysis, they are uniform.

Solution of Hyperbolic Equations Involving Chemical Reactions

Motivated by a problem in foam generation and propagation in porous media, we develop necessary and sufficient conditions for a more general quasilinear hyperbolic problem, involving reactions of finite rate, to admit as solution a steady state following the initial equilibrium state. It is found that for such a solution to exist (and to be physically well-posed), it is necessary that the generalized flux vector F and the generalized concentration vector C be colinear, namely that F = A(C)C, where A(C) is a scalar function. If an additional condition for the overlap of characteristics is valid, the two conditions are sufficient and necessary.

Solution of quasilinear hyperbolic initial-boundary-value problems by the method of pseudolinear equations

A method, called the method of pseudolinear equations, for solving a class of quasilinear hyperbolic initial-boundary-value problems of second order in one space dimension is investigated. The finite-difference implementation of the method is improved vs. a previous version by elimination of iteration. The results of a series of numerical experiments show the improved finite-difference implementation to be between 1.8 and 13.9 times faster than solution of nonlinear systems of finite-difference approximations of the quasilinear hyperbolic problems by Newton's method. However, in cases with large nonlinearities, the method of pseudolinear equations shows instability, while Newton's method converges after a large amount of computing time.