Convergence of Finite-Difference Schemes for Elliptic Equations with Variable Coefficients

We study the convergence of finite-difference schemes for second-orderelliptic equations with variable coefficients. We prove thatthe convergence rate in the discrete W₂¹ norm is of the orderh^{s –1} if the solution of the boundary value problem belongsto the Sobolev space W₂^s (1 < s 3).     

一类变系数微分方程差分格式的收敛性(英文)

本文首先对一类变系数微分方程建立有限差分格式.然后利用矩阵的特征值和范数理论,讨论该格式解的收敛性和唯一性.通过数值算例,说明该格式既有效又便于模拟.并且文中所用方法还能用于高阶微分方程和某些非线性微分方程问题的研究.     

Boundary-Value Problem for Weakly Nonlinear Hyperbolic Equations with Variable Coefficients

We establish conditions for the unique solvability of a boundary-value problem for a weakly nonlinear hyperbolic equation of order 2n, n > (3p + 1)/2, with coefficients dependent on the space coordinates and data given on the entire boundary of a cylindric domain . The investigation of this problem is connected with the problem of small denominators.     

The Rate of Convergence of Finite-Difference Approximations for Bellman Equations with Lipschitz Coefficients

We consider parabolic Bellman equations with Lipschitz coefficients. Error bounds of order h^1/2 for certain types of finite-difference schemes are obtained.     

A Dimensional Splitting Method for Quasilinear Hyperbolic Equations with Variable Coefficients

A numerical method is presented for the variable coefficient, nonlinear hyperbolic equation u _t + _i=1 ^d V _i(x, t)f _i(u) _{x i} = 0 in arbitrary space dimension for bounded velocities that are Lipschitz continuous in the x variable. The method is based on dimensional splitting and uses a recent front tracking method to solve the resulting one-dimensional non-conservative equations. The method is unconditionally stable, and it produces a subsequence that converges to the entropy solution as the discretization of time and space tends to zero. Four numerical examples are presented; numerical error mechanisms are illustrated for two linear equations, the efficiency of the method compared with a high-resolution TVD method is discussed for a nonlinear problem, and finally, applications to reservoir simulation are presented.     

Well-Posedness in Sobolev Spaces for Second-Order Strictly Hyperbolic Equations with Nondifferentiable Oscillating Coefficients

The goal of this paper is to study well-posedness to strictly hyperbolic Cauchyproblems with non-Lipschitz coefficients with low regularity with respect to timeand smooth dependence with respect to space variables. The non-Lipschitz conditionis described by the behaviour of the time-derivative of coefficients. This leads to a classification of oscillations, which has a strong influence on the loss of derivatives. To study well-posednesswe propose a refined regularizing technique. Two steps of diagonalizationprocedure basing on suitable zones of the phase spaceand corresponding nonstandard symbol classes allow to applya transformation corresponding to the effect of loss of derivatives.Finally, the application of sharp Gårding's inequality allows to derive a suitable energy estimate. From this estimatewe conclude a result about C well-posedness of the Cauchy problem.     

Hyperbolic Equations with Non-analytic Coefficients

We prove that the Cauchy problem for a hyperbolic, homogeneous equation with coefficients depending on time, is well posed in every Gevrey class, although in general it is not well-posed in provided the characteristic roots satisfy the condition

Carleman Estimates for Second-Order Hyperbolic Equations

In the space of variables (x, t) ∈ ? ⁿ⁺¹, we consider a linear second-order hyperbolic equation with coefficients depending only on x. Given a domain D ? ? ⁿ⁺¹ whose projection to the x-space is a compact domain Ω, we consider the question of construction of a stability estimate for a solution to the Cauchy problem with data on the lateral boundary S of D. The well-known method for obtaining such estimates bases on the Carleman estimates with an exponential-type weight function exp(2τ?(x, t)) whose construction faces certain difficulties in case of hyperbolic equations with variable coefficients. We demonstrate that if D is symmetric with respect to the plane t = 0 then we can take ?(x, t) to be the function ?(x, t) = s ²(x, x ⁰) ? pt ², where s(x, x ⁰) is the distance between points x and x ⁰ in the Riemannian metric induced by the differential equation, p is some positive number less than 1, and the fixed point x ⁰ can either belong to the domain Ω or lie beyond it. As for the metric, we suppose that the sectional curvature of the corresponding Riemannian space is bounded above by some number k ₀ ≥ 0. In case of space of nonpositive curvature the parameter p can be taken arbitrarily close to 1; in this case as p → 1 the stability estimates lead to a uniqueness theorem which describes exactly the domain of the solution continuation through S. It turns out that, in case of space of bounded positive curvature, construction of a Carleman estimate is possible only if the product of k ₀ and sup _x∈Ω s ²(x, x ⁰) satisfies some smallness condition.     

Oscillation of Solutions of Neutral Nonlinear Hyperbolic Equations with Doubled Variable Coefficients

In this paper, some sufficient conditions for oscillation of solutions of neutral nonlinear hyperbolic equations with doubled narialbe coefficients are obtained.These results are illustrated by some examples.     

The Rate of Convergence of Finite-Difference Approximations for Parabolic Bellman Equations with Lipschitz Coefficients in Cylindrical Domains

We consider degenerate parabolic and elliptic fully nonlinear Bellman equations with Lipschitz coefficients in domains. Error bounds of order h^1/2 in the sup norm for certain types of finite-difference schemes are obtained.     

High Accuracy A.D.I. Methods for Hyperbolic Equations with Variable Coefficients

High accuracy alternating direction implicit (A.D.I.) methodsare derived for solving the wave equation with variable coefficientsin one, two, and three space dimensions. Splittings are discussedand numerical results presented.     

Stability of Numerical Methods for Solving Second-Order Hyperbolic Equations with a Small Parameter

Doklady Mathematics - We study a symmetric three-level (in time) method with a weight and a symmetric vector two-level method for solving the initial-boundary value problem for a second-order...     

Compact Difference Schemes on a Three-Point Stencil for Second-Order Hyperbolic Equations

Differential Equations - We consider compact difference schemes of approximation order $$4+2 $$ on a three-point spatial stencil for the Klein–Gordon equations with constant and variable...     

A Remark About Hyperbolic Equations with Log-Lipschitz Coefficients

We prove the well-posedness of the Cauchy problem for strictly hyperbolic equations and systems with Log-Lipschitz coefficients in the time variable.     

Hyperbolic Equations with Growing Coefficients in Unbounded Domains

Let Ω ? ? ⁿ , n?≥?2, be an unbounded domain with a smooth (possibly noncompact) star-shaped boundary Γ. For the first mixed problem for a hyperbolic equation with an unbounded coefficient with power growth at infinity, the large-time behavior of the solutions is studied. Estimates for the resolvent of the spectral problem are obtained for various values of the parameters.     

Optimality Conditions for Semilinear Hyperbolic Equations with Controls in Coefficients

An optimal control problem for semilinear hyperbolic partial differential equations is considered. The control variable appears in coefficients. Necessary conditions for optimal controls are established by method of two-scale convergence and homogenized spike variation. Results for problems with state constraints are also stated.