Linearly implicit methods for nonlinear parabolic equations

We construct and analyze combinations of rational implicit and explicit multistep methods for nonlinear parabolic equations. The resulting schemes are linearly implicit and include as particular cases implicit-explicit multistep schemes as well as the combination of implicit Runge-Kutta schemes and extrapolation. An optimal condition for the stability constant is derived under which the schemes are locally stable. We establish optimal order error estimates.

     

Hydrodynamic limit for ∇φ interface model on a wall

 We consider random evolution of an interface on a hard wall under periodic boundary conditions. The dynamics are governed by a system of stochastic differential equations of Skorohod type, which is Langevin equation associated with massless Hamiltonian added a strong repelling force for the interface to stay over the wall. We study its macroscopic behavior under a suitable large scale space-time limit and derive a nonlinear partial differential equation, which describes the mean curvature motion except for some anisotropy effects, with reflection at the wall. Such equation is characterized by an evolutionary variational inequality. Received: 10 January 2002 / Revised version: 18 August 2002 / Published online: 15 April 2003 Mathematics Subject Classification (2000): 60K35, 82C24, 35K55, 35K85 Key words or phrases: Hydrodynamic limit – Effective interfaces – Hard wall – Skorohod's stochastic differential equation – Evolutionary variational inequality     

Backward SDEs and Cauchy problem for semilinear equations in divergence form

 We extend the definition of solutions of backward stochastic differential equations to the case where the driving process is a diffusion corresponding to symmetric uniformly elliptic divergence form operator. We show existence and uniqueness of solutions of such equations under natural assumptions on the data and show its connections with solutions of semilinear parabolic partial differential equations in Sobolev spaces. Received: 22 January 2002 / Revised version: 10 September 2002 / Published online: 19 December 2002 Research supported by KBN Grant 0253 P03 2000 19. Mathematics Subject Classification (2002): Primary 60H30; Secondary 35K55 Key words or phrases: Backward stochastic differential equation – Semilinear partial differential equation – Divergence form operator – Weak solution     

Linearly Implicit Finite Element Methods for the Time-Dependent Joule Heating Problem

Completely discrete numerical methods for a nonlinear elliptic-parabolic system, the time-dependent Joule heating problem, are introduced and analyzed. The equations are discretized in space by a standard finite element method, and in time by combinations of rational implicit and explicit multistep schemes. The schemes are linearly implicit in the sense that they require, at each time level, the solution of linear systems of equations. Optimal order error estimates are proved under the assumption of sufficiently regular solutions. AMS subject classification (2000) 65M30, 65M15, 35K60     

Local behavior of an iterative framework for generalized equations with nonisolated solutions

 An iterative framework for solving generalized equations with nonisolated solutions is presented. For generalized equations with the structure , where is a multifunction and F is single-valued, the framework covers methods that, at each step, solve subproblems of the type . The multifunction approximates F around s. Besides a condition on the quality of this approximation, two other basic assumptions are employed to show Q-superlinear or Q-quadratic convergence of the iterates to a solution. A key assumption is the upper Lipschitz-continuity of the solution set map of the perturbed generalized equation . Moreover, the solvability of the subproblems is required. Conditions that ensure these assumptions are discussed in general and by means of several applications. They include monotone mixed complementarity problems, Karush-Kuhn-Tucker systems arising from nonlinear programs, and nonlinear equations. Particular results deal with error bounds and upper Lipschitz-continuity properties for these problems. Received: November 2001 / Accepted: November 2002 Published online: December 9, 2002 Key Words. generalized equation – nonisolated solutions – Newton's method – superlinear convergence – upper Lipschitz-continuity – mixed complementarity problem – error bounds Mathematics Subject Classification (1991): 90C30, 65K05, 90C31, 90C33     

Renormalization and homogenization of fractional diffusion equations with random data

 This paper presents a renormalization and homogenization theory for fractional-in-space or in-time diffusion equations with singular random initial conditions. The spectral representations for the solutions of these equations are provided. Gaussian and non-Gaussian limiting distributions of the renormalized solutions of these equations are then described in terms of multiple stochastic integral representations. Received: 30 May 2000 / Revised version: 9 November 2001 / Published online: 10 September 2002 Mathematics Subject Classification (2000): Primary 62M40, 62M15; Secondary 60H05, 60G60 Key words or phrases: Fractional diffusion equation – Scaling laws – Renormalised solution – Long-range dependence – Non-Gaussian scenario – Mittag-Leffler function – Stable distributions – Bessel potential – Riesz potential     

Numerical integrators for quantum dynamics close to the adiabatic limit

Summary.  We study the numerical solution of singularly perturbed Schr?-dinger equations with time-dependent Hamiltonian. Based on a reformulation of the equations, we derive time-reversible numerical integrators which can be used with step sizes that are substantially larger than with traditional integration schemes. A complete error analysis is given for the adiabatic case. To deal with avoided crossings of energy levels, which lead to non-adiabatic behaviour, we propose an adaptive extension of the methods which resolves the sharp transients that appear in non-adiabatic state transitions. Received November 12, 2001 / Revised version received May 8, 2002 / Published online October 29, 2002 Mathematics Subject Classification (1991): 65L05, 65M15, 65M20, 65L70.     

A Class of Explicit Exponential General Linear Methods

In this paper, we consider a class of explicit exponential integrators that includes as special cases the explicit exponential Runge–Kutta and exponential Adams–Bashforth methods. The additional freedom in the choice of the numerical schemes allows, in an easy manner, the construction of methods of arbitrarily high order with good stability properties. We provide a convergence analysis for abstract evolution equations in Banach spaces including semilinear parabolic initial-boundary value problems and spatial discretizations thereof. From this analysis, we deduce order conditions which in turn form the basis for the construction of new schemes. Our convergence results are illustrated by numerical examples. AMS subject classification (2000) 65L05, 65L06, 65M12, 65J10     

Solving Problems with Semidefinite and Related Constraints Using Interior-Point Methods for Nonlinear Programming

 In this paper, we describe how to reformulate a problem that has second-order cone and/or semidefiniteness constraints in order to solve it using a general-purpose interior-point algorithm for nonlinear programming. The resulting problems are smooth and convex, and numerical results from the DIMACS Implementation Challenge problems and SDPLib are provided. Received: March 10, 2001 / Accepted: January 18, 2002 Published online: September 27, 2002 Key Words. semidefinite programming – second-order cone programming – interior-point methods – nonlinear programming Mathematics Subject Classification (2000): 20E28, 20G40, 20C20     

Implicit‐explicit multistep finite element methods for nonlinear convection‐diffusion problems

Implicit‐explicit multistep finite element methods for nonlinear convection‐diffusion equations are presented and analyzed. In space we discretize by finite element methods. The discretization in time is based on linear multistep schemes. The linear part of the equation is discretized implicitly and the nonlinear part of the equation explicitly. The schemes are stable and very efficient. We derive optimal order error estimates. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:93–104, 2001     

The Selfconsistent Pauli Equation

 We deal with consistent first order non-relativistic corrections (i.e. in the small parameter , where c is the speed of light) of the Dirac–Maxwell system. We discuss a selfconsistent modeling of the Pauli equation as the O(ɛ) approximation of the Dirac equation. We suggest a coupling to the “magnetostatic”O(ɛ) approximation of the Maxwell equations consisting of Poisson equations for the four components of the potential. We sketch the semiclassical/nonrelativistic limits of this model. (Received 22 May 2000)     

Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows

Summary.  We propose and analyze a semi-discrete (in time) scheme and a fully discrete scheme for the Allen-Cahn equation u _t −Δu+ɛ⁻² f(u)=0 arising from phase transition in materials science, where ɛ is a small parameter known as an ``interaction length'. The primary goal of this paper is to establish some useful a priori error estimates for the proposed numerical methods, in particular, by focusing on the dependence of the error bounds on ɛ. Optimal order and quasi-optimal order error bounds are shown for the semi-discrete and fully discrete schemes under different constraints on the mesh size h and the time step size k and different regularity assumptions on the initial datum function u ₀ . In particular, all our error bounds depend on only in some lower polynomial order for small ɛ. The cruxes of the analysis are to establish stability estimates for the discrete solutions, to use a spectrum estimate result of de Mottoni and Schatzman [18, 19] and Chen [12] and to establish a discrete counterpart of it for a linearized Allen-Cahn operator to handle the nonlinear term. Finally, as a nontrivial byproduct, the error estimates are used to establish convergence and rate of convergence of the zero level set of the fully discrete solution to the motion by mean curvature flow and to the generalized motion by mean curvature flow. Received April 30, 2001 / Revised version received March 20, 2002 / Published online July 18, 2002 Mathematics Subject Classification (1991): 65M60, 65M12, 65M15, 35B25, 35K57, 35Q99, 53A10 Correspondence to: A. Prohl     

Total Variation Diminishing Implicit Runge-Kutta Methods for Dissipative Advection-Diffusion Problems in Astrophysics

We investigate the properties of dissipative full discretizations for the equations of motion associated with models of flow and radiative transport inside stars. We derive dissipative space discretizations and demonstrate that together with specially adapted total-variation-diminishing (TVD) or strongly stable Runge-Kutta time discretizations with adaptive step-size control this yields reliable and efficient integrators for the underlying high-dimensional nonlinear evolution equations. For the most general problem class, fully implicit SDIRK methods are demonstrated to be competitive when compared to popular explicit Runge-Kutta schemes as the additional effort for the solution of the associated nonlinear equations is compensated by the larger step-sizes admissible for strong stability and dissipativity. For the parameter regime associated with semiconvection we can use partitioned IMEX Runge-Kutta schemes, where the solution of the implicit part can be reduced to the solution of an elliptic problem. This yields a significant gain in performance as compared to either fully implicit or explicit time integrators. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multi-parameter regularization techniques for ill-conditioned linear systems

Summary.  When a system of linear equations is ill-conditioned, regularization techniques provide a quite useful tool for trying to overcome the numerical inherent difficulties: the ill-conditioned system is replaced by another one whose solution depends on a regularization term formed by a scalar and a matrix which are to be chosen. In this paper, we consider the case of several regularizations terms added simultaneously, thus overcoming the problem of the best choice of the regularization matrix. The error of this procedure is analyzed and numerical results prove its efficiency. Received January 15, 2002 / Revised version received July 31, 2002 / Published online October 29, 2002 Mathematics Subject Classification (1991): 65F05 – 65F22     

First- and second-order methods for semidefinite programming

 In this paper, we survey the most recent methods that have been developed for the solution of semidefinite programs. We first concentrate on the methods that have been primarily motivated by the interior point (IP) algorithms for linear programming, putting special emphasis in the class of primal-dual path-following algorithms. We also survey methods that have been developed for solving large-scale SDP problems. These include first-order nonlinear programming (NLP) methods and more specialized path-following IP methods which use the (preconditioned) conjugate gradient or residual scheme to compute the Newton direction and the notion of matrix completion to exploit data sparsity. Received: December 16, 2002 / Accepted: May 5, 2003 Published online: May 28, 2003 Key words. semidefinite programming – interior-point methods – polynomial complexity – path-following methods – primal-dual methods – nonlinear programming – Newton method – first-order methods – bundle method – matrix completion The author's research presented in this survey article has been supported in part by NSF through grants INT-9600343, INT-9910084, CCR-9700448, CCR-9902010, CCR-0203113 and ONR through grants N00014-93-1-0234, N00014-94-1-0340 and N00014-03-1-0401. Mathematics Subject Classification (2000): 65K05, 90C06, 90C22, 90C25, 90C30, 90C51     

Numerical schemes for Hamilton-Jacobi equations on unstructured meshes

Summary.  In this paper, a numerical scheme is presented by applying the finite element method to the viscosity equations of the Hamilton-Jacobi equations on unstructured meshes. By improving the finite element scheme, another numerical scheme is constructed. Under certain limitations, the numerical solutions of the two schemes converge to the viscosity solutions of the Hamilton-Jacobi equations. The latter numerical scheme has weaker restrictions than the former scheme for convergence. Numerical examples are provided to test the stability, convergence and sensitivity to different meshes. Received November 5, 2001 / Revised version received March 5, 2002 / Published online October 29, 2002 RID="*" ID="*" Current address: Department of Applied Mathematics, University of Petroleum, Dongying 257062, Shandong, P.R.China; e-mail: xianggui_li@sina.com Mathematics Subject Classification (1991): 65M60     

Applications of Lobatto polynomials to an adaptive finite element method: A posteriori error estimates for HP-Adaptivity and Grid-to-Grid Interpolation

Summary.  Hp-adaptive finite element codes require methods for estimating the error at several spatial orders and for interpolating solutions between grids. Lobatto polynomial-based techniques are presented for both. An interpolation error-based error estimation strategy for a posteriori error estimates is generalized to yield asymptotically exact error estimates one order higher than the computed solution. The estimates involve high-order derivatives of the solution that must be approximated from the computed solution. Differentiating a ``Taylor-like' series for error in the Lobatto interpolant and using the weak form of the equations yields the correct derivative approximations. This leads to a more robust order selection strategy. Interpolation between grids is done over each element using the Lobatto interpolating polynomial. Explicit formulas for the inverse of the resulting Lobatto interpolation matrices are given. Computational results illustrate the theory. Received June 25, 2001 / Revised version received February 12, 2002 / Published online October 29, 2002 Mathematics Subject Classification (1991): 65M15,65M20,65M60 This research was partially supported by NSF Grant #DMS-0196108.     

Anisotropic error estimates for elliptic problems

Summary.  Moving from the anisotropic interpolation error estimates derived in [12], we provide here both a-priori and a-posteriori estimates for a generic elliptic problem. The a-priori result is deduced by following the standard finite element theory. For the a-posteriori estimate, the analysis extends to anisotropic meshes the theory presented in [3–5]. Numerical test-cases validate the derived results. Received July 22, 2001 / Revised version received March 20, 2002 / Published online July 18, 2002 Mathematics Subject Classification (1991): 65N15, 65N50     

The Selfconsistent Pauli Equation

 We deal with consistent first order non-relativistic corrections (i.e. in the small parameter , where c is the speed of light) of the Dirac–Maxwell system. We discuss a selfconsistent modeling of the Pauli equation as the O(ɛ) approximation of the Dirac equation. We suggest a coupling to the “magnetostatic”O(ɛ) approximation of the Maxwell equations consisting of Poisson equations for the four components of the potential. We sketch the semiclassical/nonrelativistic limits of this model.     

A semi-discrete scheme for the stochastic nonlinear Schrödinger equation

Summary. We study the convergence of a semi-discretized version of a numerical scheme for a stochastic nonlinear Schrödinger equation. The nonlinear term is a power law and the noise is multiplicative with a Stratonovich product. Our scheme is implicit in the deterministic part of the equation as is usual for conservative equations. We also use an implicit discretization of the noise which is better suited to Stratonovich products. We consider a subcritical nonlinearity so that the energy can be used to obtain an a priori estimate. However, in the semi discrete case, no Ito formula is available and we have to use a discrete form of this tool. Also, in the course of the proof we need to introduce a cut-off of the diffusion coefficient, which allows to treat the nonlinearity. Then, we prove convergence by a compactness argument. Due to the presence of noise and to the implicit discretization of the noise, this is rather complicated and technical. We finally obtain convergence of the discrete solutions in various topologies. Mathematics Subject Classification (2000):35Q55, 60H15, 65M06, 65M12