An operator splitting method for nonlinear convection-diffusion equations

Summary. We present a semi-discrete method for constructing approximate solutions to the initial value problem for the -dimensional convection-diffusion equation . The method is based on the use of operator splitting to isolate the convection part and the diffusion part of the equation. In the case , dimensional splitting is used to reduce the -dimensional convection problem to a series of one-dimensional problems. We show that the method produces a compact sequence of approximate solutions which converges to the exact solution. Finally, a fully discrete method is analyzed, and demonstrated in the case of one and two space dimensions. ReceivedFebruary 1, 1996 / Revised version received June 24, 1996     

Parameter estimation in systems of nonlinear equations

Summary. A numerical method is presented to determine parameters in a system of non\-linear equations in the following sense: Proceeding from given experimental data, e.g., observation times and measurements, the minimum least-squares distance of the measured data from a fitting criterion depending on the solution of a system of nonlinear equations is to be computed. Specifically coupled mass equilibrium models are described in more detail that are used in pharmaceutical applications for receptor-ligand binding studies. They are used for instance for the radioimmunological determination of Fenoterol or related substances. Numerical results based on laboratory data are included to test the robustness of the algorithm implemented. Received May 24, 1993/Revised version received February 13, 1994     

The parity of the Zeckendorf sum-of-digits function

Let s(n) denote the sum of digits of the Zeckendorf representation of n and . The aim of this paper is to discuss the behaviour of $S_{q,i}(N)$. First it is shown that that the values of admit a Gaussian limit law with bounded mean and variance of order log N. Conversely, for q≤1 (mostly) has a periodic fractal structure. We also prove that which is an analogue to a well-known result by Newman [14] for binary digit expansions. Received: 15 March 1999 / Revised version: 5 November 1999     

Concavity estimates for a class of nonlinear elliptic equations in two dimensions

We study solutions of the nonlinear elliptic equation on a bounded domain in . It is shown that the set of points where the graph of the solution has negative Gauss curvature always extends to the boundary, unless it is empty. The meethod uses an elliptic equation satisfied by an auxiliary function given by the product of the Hessian determinant and a suitable power of the solutions. As a consequence of the result, we give a new proof for power concavity of solutions to certain semilinear boundary value problems in convex domains. Received: 12 January 2000; in final form: 15 March 2001 / Published online: 4 April 2002     

Asymptotic stability analysis of Runge-Kutta methods for nonlinear systems of delay differential equations

Summary. We consider systems of delay differential equations (DDEs) of the form with the initial condition . Recently, Torelli [10] introduced a concept of stability for numerical methods applied to dissipative nonlinear systems of DDEs (in some inner product norm), namely RN-stability, which is the straighforward generalization of the wellknown concept of BN-stability of numerical methods with respect to dissipative systems of ODEs. Dissipativity means that the solutions and corresponding to different initial functions and , respectively, satisfy the inequality , and is guaranteed by suitable conditions on the Lipschitz constants of the right-hand side function . A numerical method is said to be RN-stable if it preserves this contractivity property. After showing that, under slightly more stringent hypotheses on the Lipschitz constants and on the delay function , the solutions and are such that , in this paper we prove that RN-stable continuous Runge-Kutta methods preserve also this asymptotic stability property. Received March 29, 1996 / Revised version received August 12, 1996     

Rational approximation to orthogonal bases and of the solutions of elliptic equations

It is shown that given , an orthogonal basis of can be approximated by an orthogonal basis , where has integral and have rational components, such that the angle between and is at most and the length , . This improves the length of the integral approximation due to Schmidt (1995). As an application, we improve a theorem of Kocan (1995) about the minimal size of grids in the solutions of elliptic equations. Our result fits the need in Kuo and Trudinger (1990). Received January 27, 1997 / Revised version received April 1, 1998     

Convergent difference schemes for nonlinear parabolic equations and mean curvature motion

Summary. Explicit finite difference schemes are given for a collection of parabolic equations which may have all of the following complex features: degeneracy, quasilinearity, full nonlinearity, and singularities. In particular, the equation of “motion by mean curvature” is included. The schemes are monotone and consistent, so that convergence is guaranteed by the general theory of approximation of viscosity solutions of fully nonlinear problems. In addition, an intriguing new type of nonlocal problem is analyzed which is related to the schemes, and another very different sort of approximation is presented as well. Received January 10, 1995     

Regular solutions of nonlinear differential-algebraic equations and their numerical determination

For a general class of nonlinear (possibly higher index) differential-algebraic equations we show existence and uniqueness of solutions. These solutions are regular in the sense that Newton's method will converge locally and quadratically. On the basis of the presented theoretical results, numerical methods for the determination of consistent initial values and for the computation of regular solutions are developed. Several numerical examples are included. Received July 4, 1996 / Revised version received May 12, 1997     

Volterra discrete equations: summability of the fundamental matrix

Summary. Conditions are proven which assure the summability of the first difference of the fundamental matrix of nonconvolution Volterra discrete equations. These conditions are applied to the stability analysis of some linear methods for solving Volterra integral equations of nonconvolution type. Received July 25, 1999 / Revised version received February 14, 2000 / Published online April 5, 2001     

Accelerated monotone iterative methods for finite difference equations of reaction-diffusion

This paper is concerned with numerical methods for a finite difference system of reaction-diffusion-convection equation under nonlinear boundary condition. Various monotone iterative methods are presented, and each of these methods leads to an existence-comparison theorem as well as a computational algorithm for numerical solutions. The monotone property of the iterations gives improved upper and lower bounds of the solution in each iteration, and the rate of convergence of the iterations is either quadratic or nearly quadratic depending on the property of the nonlinear function. Application is given to a model problem from chemical engineering, and some numerical results, including a test problem with known analytical solution, are presented to illustrate the various rates of convergence of the iterations. Received November 2, 1995 / Revised version received February 10, 1997     

On the stability of the unsmoothed Fourier method for hyperbolic equations

Summary. It has been a long open question whether the pseudospectral Fourier method without smoothing is stable for hyperbolic equations with variable coefficients that change signs. In this work we answer this question with a detailed stability analysis of prototype cases of the Fourier method. We show that due to weighted -stability, the -degree Fourier solution is algebraically stable in the sense that its amplification does not exceed . Yet, the Fourier method is weakly -unstable in the sense that it does experience such amplification. The exact mechanism of this weak instability is due the aliasing phenomenon, which is responsible for an amplification of the Fourier modes at the boundaries of the computed spectrum. Two practical conclusions emerge from our discussion. First, the Fourier method is required to have sufficiently many modes in order to resolve the underlying phenomenon. Otherwise, the lack of resolution will excite the weak instability which will propagate from the slowly decaying high modes to the lower ones. Second -- independent of whether smoothing was used or not, the small scale information contained in the highest modes of the Fourier solution will be destroyed by their amplification. Happily, with enough resolution nothing worse can happen. Received December 14, 1992/Revised version received March 1, 1993     

A stable and conservative finite difference scheme for the Cahn-Hilliard equation

Summary. We propose a stable and conservative finite difference scheme to solve numerically the Cahn-Hilliard equation which describes a phase separation phenomenon. Numerical solutions to the equation is hard to obtain because it is a nonlinear and nearly ill-posed problem. We design a new difference scheme based on a general strategy proposed recently by Furihata and Mori. The new scheme inherits characteristic properties, the conservation of mass and the decrease of the total energy, from the equation. The decrease of the total energy implies boundedness of discretized Sobolev norm of the solution. This in turn implies, by discretized Sobolev's lemma, boundedness of max norm of the solution, and hence the stability of the solution. An error estimate for the solution is obtained and the order is . Numerical examples demonstrate the effectiveness of the proposed scheme. Received July 22, 1997 / Revised version received October 19, 1999 / Published online August 2, 2000     

Absorbing boundary conditions for diffusion equations

Summary. We construct and analyse a family of absorbing boundary conditions for diffusion equations with variable coefficients, curved artifical boundary, and arbitrary convection. It relies on the geometric identification of the Dirichlet to Neumann map and rational interpolation of in the complex plane. The boundary conditions are stable, accurate, and practical for computations. Received December 12, 1992 / Revised version received July 4, 1994     

Analysis of finite difference schemes for unsteady Navier-Stokes equations in vorticity formulation

Summary. In this paper, we provide stability and convergence analysis for a class of finite difference schemes for unsteady incompressible Navier-Stokes equations in vorticity-stream function formulation. The no-slip boundary condition for the velocity is converted into local vorticity boundary conditions. Thom's formula, Wilkes' formula, or other local formulas in the earlier literature can be used in the second order method; while high order formulas, such as Briley's formula, can be used in the fourth order compact difference scheme proposed by E and Liu. The stability analysis of these long-stencil formulas cannot be directly derived from straightforward manipulations since more than one interior point is involved in the formula. The main idea of the stability analysis is to control local terms by global quantities via discrete elliptic regularity for stream function. We choose to analyze the second order scheme with Wilkes' formula in detail. In this case, we can avoid the complicated technique necessitated by the Strang-type high order expansions. As a consequence, our analysis results in almost optimal regularity assumption for the exact solution. The above methodology is very general. We also give a detailed analysis for the fourth order scheme using a 1-D Stokes model. Received December 10, 1999 / Revised version received November 5, 2000 / Published online August 17, 2001     

Block monotone iterative methods for numerical solutions of nonlinear elliptic equations

Summary. Two block monotone iterative schemes for a nonlinear algebraic system, which is a finite difference approximation of a nonlinear elliptic boundary-value problem, are presented and are shown to converge monotonically either from above or from below to a solution of the system. This monotone convergence result yields a computational algorithm for numerical solutions as well as an existence-comparison theorem of the system, including a sufficient condition for the uniqueness of the solution. An advantage of the block iterative schemes is that the Thomas algorithm can be used to compute numerical solutions of the sequence of iterations in the same fashion as for one-dimensional problems. The block iterative schemes are compared with the point monotone iterative schemes of Picard, Jacobi and Gauss-Seidel, and various theoretical comparison results among these monotone iterative schemes are given. These comparison results demonstrate that the sequence of iterations from the block iterative schemes converges faster than the corresponding sequence given by the point iterative schemes. Application of the iterative schemes is given to a logistic model problem in ecology and numerical ressults for a test problem with known analytical solution are given. Received August 1, 1993 / Revised version received November 7, 1994     

Convergence and stability of finite element nonlinear Galerkin method for the Navier-Stokes equations

In this article, we present a new fully discrete finite element nonlinear Galerkin method, which are well suited to the long time integration of the Navier-Stokes equations. Spatial discretization is based on two-grid finite element technique; time discretization is based on Euler explicit scheme with variable time step size. Moreover, we analyse the boundedness, convergence and stability condition of the finite element nonlinear Galerkin method. Our discussion shows that the time step constraints of the method depend only on the coarse grid parameter and the time step constraints of the finite element Galerkin method depend on the fine grid parameter under the same convergence accuracy. Received February 2, 1994 / Revised version received December 6, 1996     

Preserving monotonicity in the numerical solution of Riccati differential equations

Summary. Solutions of symmetric Riccati differential equations (RDEs for short) are in the usual applications positive semidefinite matrices. Moreover, in the class of semidefinite matrices, solutions of different RDEs are also monotone, with respect to properly ordered data. Positivity and monotonicity are essential properties of RDEs. In Dieci and Eirola (1994), we showed that, generally, a direct discretization of the RDE cannot maintain positivity, and be of order greater than one. To get higher order, and to maintain positivity, we are thus forced to look into indirect solution procedures. Here, we consider the problem of how to maintain monotonicity in the numerical solutions of RDEs. Naturally, to obtain order greater than one, we are again forced to look into indirect solution procedures. Still, the restrictions imposed by monotonicity are more stringent that those of positivity, and not all of the successful indirect solution procedures of Dieci and Eirola (1994) maintain monotonicity. We prove that by using symplectic Runge-Kutta (RK) schemes with positive weights (e.g., Gauss schemes) on the underlying Hamiltonian matrix, we eventually maintain monotonicity in the computed solutions of RDEs. Received May 2, 1995