Convergence of a finite volume scheme for nonlinear degenerate parabolic equations

Summary. One approximates the entropy weak solution u of a nonlinear parabolic degenerate equation by a piecewise constant function using a discretization in space and time and a finite volume scheme. The convergence of to u is shown as the size of the space and time steps tend to zero. In a first step, estimates on are used to prove the convergence, up to a subsequence, of to a measure valued entropy solution (called here an entropy process solution). A result of uniqueness of the entropy process solution is proved, yielding the strong convergence of to{\it u}. Some on a model equation are shown. Received September 27, 2000 / Published online October 17, 2001     

Averaging the trajectory attractor of a nonlinear wave equation with rapidly oscillating right-hand side

We consider a nonlinear nonautonomous hyperbolic equation with dissipation and with a small parameter multiplying the highest derivative with respect to time. This equation also involves a rapidly oscillating external force. Using a standard technique for constructing the trajectory attractor, we can prove the convergence of the attractor of a nonlinear nonautonomous hyperbolic equation with dissipation to the attractor of the corresponding parabolic equation.     

Spectral methods based on Hermite functions for linear hyperbolic equations

We consider the approximation by spectral and pseudo‐spectral methods of the solution of the Cauchy problem for a scalar linear hyperbolic equation in one space dimension posed in the whole real line. To deal with the fact that the domain of the equation is unbounded, we use Hermite functions as orthogonal basis. Under certain conditions on the coefficients of the equation, we prove the spectral convergence rate of the approximate solutions for regular initial data in a weighted space related to the Hermite functions. Numerical evidence of this convergence is also presented. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012     

Steady transonic shocks and free boundary problems in infinite cylinders for the Euler equations

We establish the existence and stability of multidimensional transonic shocks (hyperbolic‐elliptic shocks) for the Euler equations for steady compressible potential fluids in infinite cylinders. The Euler equations, consisting of the conservation law of mass and the Bernoulli law for velocity, can be written as a second order nonlinear equation of mixed elliptic‐hyperbolic type for the velocity potential. The transonic shock problem in an infinite cylinder can be formulated into the following free boundary problem: The free boundary is the location of the multidimensional transonic shock which divides two regions of C^1,α flow in the infinite cylinder, and the equation is hyperbolic in the upstream region where the C^1,α perturbed flow is supersonic. We develop a nonlinear approach to deal with such a free boundary problem in order to solve the transonic shock problem in unbounded domains. Our results indicate that there exists a solution of the free boundary problem such that the equation is always elliptic in the unbounded downstream region, the uniform velocity state at infinity in the downstream direction is uniquely determined by the given hyperbolic phase, and the free boundary is C^1,α, provided that the hyperbolic phase is close in C^1,α to a uniform flow. We further prove that, if the steady perturbation of the hyperbolic phase is C^2,α, the free boundary is C^2,α and stable under the steady perturbation. © 2003 Wiley Periodicals Inc.     

On the numerical analysis of finite element and dirichlet-to-neumann methods for nonlinear exterior transmission problems

We provide the numerical analysis of the combination of finite elements and Dirichlet-to-Neumann mappings (based on boundary integral operators) for a class of nonlinear exterior transmission problems whose weak formulations reduce to Lipschitz-continuous and strongly monotone operator equations. As a model we consider a nonlinear second order elliptic equation in divergence form in a bounded inner region of the plane, coupled with the Laplace equation in the corresponding unbounded exterior part. A discrete Galerkin scheme is presented by using linear finite elements on a triangulation of the domain, and then applying numerical quadrature and analytical formulae to evaluate all the linear, bilinear and semilinear forms involved. We prove the unique solvability of the discrete equations, and show the strong convergence of the approximate solutions. Furthermore, assuming additional regularity on the solution of the continuous operator equation, the asymptotic rate of convergence O(h) is also derived. Finally, numerical experiments are presented, which confirm the convergence results.     

EXISTENCE AND UNIQUENESS OF THE ENTROPY SOLUTION TO A NONLINEAR HYPERBOLIC EQUATION

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Unboundedness of solutions of one third-order hyperbolic equation

We have established conditions under which the solution of mixed problem for one third-order nonlinear hyperbolic equation becomes unbounded at a finite moment of time.     

On the existence of generalized solutions and the convergence of difference methods for nonlinear initial-value problems

In the present paper we prove new results for a general perturbation theory for nonlinear mappings between metric spaces. Using these results we are able to establish new principles for the treatment of nonlinear initial-value problems by difference methods. The main results are the characterization of the existence of discrete limits of sequences of mappings and the characterization of the existence of generalized solutions of nonlinear initial-value problems which are limits of solutions of difference equations. As conclusions one obtains generalizations of Lax's equivalence theorem for nonlinear and linear initial-value problems and a convergence theorem for a concrete hyperbolic equation.     

Iterative method for solving an inverse coefficient problem for a hyperbolic equation

For a hyperbolic equation, we consider an inverse coefficient problem in which the unknown coefficient occurs in both the equation and the initial condition. The solution values on a given curve serve as additional information for determining the unknown coefficient. We suggest an iterative method for solving the inverse problem based on reduction to a nonlinear operator equation for the unknown coefficient and prove the uniform convergence of the iterations to a function that is a solution of the inverse problem.     

A reduced‐order extrapolated natural boundary element method based on POD for the 2D hyperbolic equation in unbounded domain

In this article, we primarily focuses to study the order‐reduction for the classical natural boundary element (NBE) method for the two‐dimensional (2D) hyperbolic equation in unbounded domain. To this end, we first build a semi‐discretized format about time for the hyperbolic equation and discuss the existence, stability, and convergence of the time semi‐discretized solutions. We then establish the classical fully discretized NBE format from the time semi‐discretized one and analyze the existence, stability, and convergence of the classical NBE solutions. Next, using proper orthogonal decomposition method, we build a reduced‐order extrapolated NBE (ROENBE) format containing very few unknowns but having adequately high accuracy, and we also discuss the existence, stability, and convergence of the ROENBE solutions. Finally, we use some numerical examples to show that the ROENBE method is far superior to the classical NBE one. It shows that the ROENBE method is reliable and effective for solving the 2D hyperbolic equation with the unbounded domain.     

Euler-Poisson-Darboux differential-operator equation with variable domains of smooth operators

We prove the strong well-posed solvability of the Cauchy problem for a second-order singular hyperbolic differential equation with variable domain of variable unbounded operator coefficients and for the mixed problem for a complete equation of string vibrations with a strong singularity in time and with a time-dependent boundary condition.     

Long-time behavior of solutions to a nonlinear hyperbolic relaxation system

We study the large time behavior of solutions of a one-dimensional hyperbolic relaxation system that may be written as a nonlinear damped wave equation. First, we prove the global existence of a unique solution and their decay properties for sufficiently small initial data. We also show that for some large initial data, solutions blow-up in finite time. For quadratic nonlinearities, we prove that the large time behavior of solutions is given by the fundamental solution of the viscous Burgers equation. In some other cases, the convection term is too weak and the large time behavior is given by the linear heat kernel.     

On solutions of a quadratic integral equation of Hammerstein type

We study the solvability of a nonlinear quadratic integral equation of Hammerstein type. Using the technique of measures of noncompactness we prove that this equation has solutions on an unbounded interval. Moreover, we also obtain an asymptotic characterization of these solutions. Several special cases of this integral equation are discussed and applications to real world problems are indicated.     

A New Class of Uniformly Second Order Accurate Difference Schemes for 2D Scalar Conservation Laws

In this paper, concerned with the Cauchy problem for 2D nonlinear hyperbolic conservation laws, we construct a class of uniformly second order accurate finite difference schemes, which are based on the E-schemes. By applying the convergence theorem of Coquel-Le Floch [1], the family of approximate solutions defined by the scheme is proven to converge to the unique entropy weak $L^{\infty}$-solution. Furthermore, some numerical experiments on the Cauchy problem for the advection equation and the Riemann problem for the 2D Burgers equation are given and the relatively satisfied result is obtained.     

A new nonlinear functional for general scalar conservation laws

The approach based on the construction of some nonlinear functionals was proved to be robust in the study of the well-posedness theories of hyperbolic conservation laws, especially in one space dimensional case. In particular, a generalized entropy functional was constructed in [T.-P. Liu, T. Yang, A new entropy functional for scalar conservation laws, Comm. Pure Appl. Math. 52 (1999) 1427-1442] for the L¹ stability of weak solutions. However, this generalized functional is so far only defined for scalar equations with convex flux function. In this paper, we introduce a new nonlinear functional which is motivated by the new Glimm functional introduced in [J.-L. Hua, Z.-H. Jiang, T. Yang, A new Glimm functional and convergence rate of Glimm scheme for general systems of hyperbolic conservation laws, preprint] for general scalar conservation laws. This functional improves the one given in [H.-X. Liu, T. Yang, A nonlinear functional for general scalar hyperbolic conservation laws, J. Differential Equations 235 (2) (2007) 658-667] and it can be viewed as a better attempt for the generalized entropy functional for general equations.     

Stabilization of the statistical solution of the parabolic equation

We study the convergence of the statistical solutions of the parabolic equation. Under some mixing condition (in the sense of Rosenblatt) for initial measure and natural assumptions on the coefficients of the equation we prove weak convergence to the Gaussian distribution. Similar results for the hyperbolic equations were obtained in [1–4].     

On Wavewise Entropy Inequalities for High-Resolution Schemes. I: The Semidiscrete Case

We develop a new approach, the method of wavewise entropy inequalities for the numerical analysis of hyperbolic conservation laws. The method is based on a new extremum tracking theory and Volpert's theory of BV solutions. The method yields a sharp convergence criterion which is used to prove the convergence of generalized MUSCL schemes and a class of schemes using flux limiters previously discussed in 1984 by Sweby.

     

Overlapping Schwarz Waveform Relaxation for the Heat Equation in N Dimensions

We analyze overlapping Schwarz waveform relaxation for the heat equation in n spatial dimensions. We prove linear convergence of the algorithm on unbounded time intervals and superlinear convergence on bounded time intervals. In both cases the convergence rates are shown to depend on the size of the overlap. The linear convergence result depends also on the number of subdomains because it is limited by the classical steady state result of overlapping Schwarz for elliptic problems. However the superlinear convergence result is independent of the number of subdomains. Thus overlapping Schwarz waveform relaxation does not need a coarse space for robust convergence independent of the number of subdomains, if the algorithm is in the superlinear convergence regime. Numerical experiments confirm our analysis. We also briefly describe how our results can be extended to more general parabolic problems.     

Approximation Spaces in the Numerical Analysis of Operator Equations

We show that generalized approximation spaces can be used to prove stability and convergence of projection methods for certain types of operator equations in which unbounded operators occur. Besides the convergence, we also get orders of convergence by this approach, even in case of non-uniformly bounded projections. We give an example in which weighted uniform convergence of the collocation method for an easy Cauchy singular integral equation is studied.     

Existence results for an indefinite unbounded perturbation of a resonant Schrödinger equation

We study existence results for a nonlinear Schrödinger equation at resonance. The nonlinearity is assumed to change sign, be unbounded but sublinear with a power like growth at infinity. Under a suitable coercivity assumption on the primitive of the nonlinear term on the kernel of the Schrödinger operator, we prove the existence of at least one solution.