Numerical differentiation for the second order derivatives of functions of two variables

A regularized optimization problem for computing numerical differentiation for the second order derivatives of functions with two variables from noisy values at scattered points is discussed in this article. We prove the existence and uniqueness of the solution to this problem, provide a constructive scheme for the solution which is based on bi-harmonic Green's function and give a convergence estimate of the regularized solution to the exact solution for the problem under a simple choice of regularization parameter. The efficiency of the constructive scheme is shown by some numerical examples.     

Existence and uniqueness for a nonlinear parabolic/Hamilton–Jacobi coupled system describing the dynamics of dislocation densities

We study a mathematical model describing the dynamics of dislocation densities in crystals. This model is expressed as a 1D system of a parabolic equation and a first order Hamilton–Jacobi equation that are coupled together. We examine an associated Dirichlet boundary value problem. We prove the existence and uniqueness of a viscosity solution among those assuming a lower-bound on their gradient for all time including the initial time. Moreover, we show the existence of a viscosity solution when we have no such restriction on the initial data. We also state a result of existence and uniqueness of entropy solution for the initial value problem of the system obtained by spatial derivation. The uniqueness of this entropy solution holds in the class of bounded-from-below solutions. In order to prove our results on the bounded domain, we use an “extension and restriction” method, and we exploit a relation between scalar conservation laws and Hamilton–Jacobi equations, mainly to get our gradient estimates.     

Solving neutral delay differential equations with state-dependent delays

In this paper we consider a class of neutral delay differential equations with state dependent delays. For such equations the possible discontinuity in the derivative of the solution at the initial point may propagate along the integration interval giving rise to subsequent points, called “breaking points”, where the solution derivative is still discontinuous. As a consequence, in a right neighbourhood of each such point we have to face a Cauchy problem where the equation has a discontinuous right-hand side. In this case the existence and the uniqueness of the solution is no longer guaranteed to the right of such points and hence the solution of the neutral equation may either cease to exist or bifurcate. After illustrating why uniqueness and existence of the solution is no longer guaranteed for general state-dependent problems and showing a possible way to detect these occurrences automatically, we explain how to generalize/regularize the problem in order to suitably extend the solution beyond the breaking point. This is important, for example, when exploring numerically the presence of possible periodic orbits.     

Algorithms for the computation of functionals defined on the solution of a discrete ill-posed problem

We study a linear, discrete ill-posed problem, by which we mean a very ill-conditioned linear least squares problem. In particular we consider the case when one is primarily interested in computing a functional defined on the solution rather than the solution itself. In order to alleviate the ill-conditioning we require the norm of the solution to be smaller than a given constant. Thus we are lead to minimizing a linear functional subject to two quadratic constraints. We study existence and uniqueness for this problem and show that it is essentially equivalent to a least squares problem with a linear and a quadratic constraint, which is easier to handle computationally. Efficient algorithms are suggested for this problem.     

Relaxation approximation to bed-load sediment transport

In this work we propose and apply a numerical method based on finite volume relaxation approximation for computing the bed-load sediment transport in shallow water flows, in one and two space dimensions. The water flow is modeled by the well-known nonlinear shallow water equations which are coupled with a bed updating equation. Using a relaxation approximation, the nonlinear set of equations (and for two different formulations) is transformed to a semilinear diagonalizable problem with linear characteristic variables. A second order MUSCL-TVD method is used for the advection stage while an implicit–explicit Runge–Kutta scheme solves the relaxation stage. The main advantages of this approach are that neither Riemann problem solvers nor nonlinear iterations are required during the solution process. For the two different formulations, the applicability and effectiveness of the presented scheme is verified by comparing numerical results obtained for several benchmark test problems.     

Conservation law with the flux function discontinuous in the space variable—II: Convex–concave type fluxes and generalized entropy solutions

We deal with a single conservation law with discontinuous convex–concave type fluxes which arise while considering sign changing flux coefficients. The main difficulty is that a weak solution may not exist as the Rankine–Hugoniot condition at the interface may not be satisfied for certain choice of the initial data. We develop the concept of generalized entropy solutions for such equations by replacing the Rankine–Hugoniot condition by a generalized Rankine–Hugoniot condition. The uniqueness of solutions is shown by proving that the generalized entropy solutions form a contractive semi-group in L¹$L^1$. Existence follows by showing that a Godunov type finite difference scheme converges to the generalized entropy solution. The scheme is based on solutions of the associated Riemann problem and is neither consistent nor conservative. The analysis developed here enables to treat the cases of fluxes having at most one extrema in the domain of definition completely. Numerical results reporting the performance of the scheme are presented.     

A convergent scheme for a non local Hamilton Jacobi equation modelling dislocation dynamics

We study dislocation dynamics with a level set point of view. The model we present here looks at the zero level set of the solution of a non local Hamilton Jacobi equation, as a dislocation in a plane of a crystal. The front has a normal speed, depending on the solution itself. We prove existence and uniqueness for short time in the set of continuous viscosity solutions. We also present a first order finite difference scheme for the corresponding level set formulation of the model. The scheme is based on monotone numerical Hamiltonian, proposed by Osher and Sethian. The non local character of the problem makes it not monotone. We obtain an explicit convergence rate of the approximate solution to the viscosity solution. We finally provide numerical simulations.This work has been supported by funds from ACI JC 1041 “Mouvements d’interfaces avec termes non-locaux”, from ACI-JC 1025 “Dynamique des dislocations” and from ONERA, Office National d’Etudes et de Recherches. The second author was also supported by the ENPC-Région Ile de France.     

Numerical analysis for a conservative difference scheme to solve the Schrödinger-Boussinesq equation

In this paper, we present a finite difference scheme for the solution of an initial-boundary value problem of the Schrödinger-Boussinesq equation. The scheme is fully implicit and conserves two invariable quantities of the system. We investigate the existence of the solution for the scheme, give computational process for the numerical solution and prove convergence of iteration method by which a nonlinear algebra system for unknown Vⁿ⁺¹ is solved. On the basis of a priori estimates for a numerical solution, the uniqueness, convergence and stability for the difference solution is discussed. Numerical experiments verify the accuracy of our method.     

An Engquist-Osher type finite difference scheme with a discontinuous flux function in space

An Engquist-Osher type finite difference scheme is derived for dealing with scalar conservation laws having a flux that is spatially dependent through a possibly discontinuous coefficient. The new monotone difference scheme is based on introducing a new interface numerical flux function, which is called a generalized Engquist-Osher flux. By means of this scheme, the existence and uniqueness of weak solutions to the scalar conservation laws are obtained and the convergence theorem is established. Some numerical examples are presented and the corresponding numerical results are displayed to illustrate the efficiency of the methods.     

A second-order difference scheme for a parameterized singular perturbation problem

In this paper we consider a singularly perturbed quasilinear boundary value problem depending on a parameter. The problem is discretized using a hybrid difference scheme on Shishkin-type meshes. We show that the scheme is second-order convergent, in the discrete maximum norm, independent of singular perturbation parameter. Numerical experiments support these theoretical results.     

Some results on condition numbers of the scaled total least squares problem

Under the Golub-Van Loan condition for the existence and uniqueness of the scaled total least squares (STLS) solution, a first order perturbation estimate for the STLS solution and upper bounds for condition numbers of a STLS problem have been derived by Zhou et al. recently. In this paper, a different perturbation analysis approach for the STLS solution is presented. The analyticity of the solution to the perturbed STLS problem is explored and a new expression for the first order perturbation estimate is derived. Based on this perturbation estimate, for some STLS problems with linear structure we further study the structured condition numbers and derive estimates for them. Numerical experiments show that the structured condition numbers can be markedly less than their unstructured counterparts.     

A two-dimensional glimm type scheme on cauchy problem of two-dimensional scalar conservation law

In this paper,we construct a new two-dimensional convergent scheme to solve Cauchy problem of following two-dimensional scalar conservation law{?_tu +?_xf(u) + ?_yg(u) = 0,u(x,y,0) = u_0(x,y).In which initial data can be unbounded.Although the existence and uniqueness of the weak entropy solution are obtained,little is known about how to investigate two-dimensional or higher dimensional conservation law by the schemes based on wave interaction of 2D Riemann solutions and their estimation.So we construct such scheme in our paper and get some new results.     

Parallel defect control

How can small-scale parallelism best be exploited in the solution of nonstiff initial value problems? It is generally accepted that only modest gains inefficiency are possible, and it is often the case that “fast” parallel algorithms have quite crude error control and stepsize selection components. In this paper we consider the possibility of using parallelism to improvereliability andfunctionality rather than efficiency. We present an algorithm that can be used with any explicit Runge-Kutta formula. The basic idea is to take several smaller substeps in parallel with the main step. The substeps provide an interpolation facility that is essentially free, and the error control strategy can then be based on a defect (residual) sample. If the number of processors exceeds (p ? 1)/2, wherep is the order of the Runge-Kutta formula, then the interpolant and the error control scheme satisfy very strong reliability conditions. Further, for a given orderp, the asymptotically optimal values for the substep lengths are independent of the problem and formula and hence can be computed a priori. Theoretical comparisons between the parallel algorithm and optimal sequential algorithms at various orders are given. We also report on numerical tests of the reliability and efficiency of the new algorithm, and give some parallel timing statistics from a 4-processor machine.     

On the approximation of nonlinear singular self-adjoint second order boundary value problems

We investigate the approximation of the solutions of a class of nonlinear second order singular boundary value problems with a self-adjoint linear part. Our strategy involves two ingredients. First, we take advantage of certain boundary condition functions to obtain well behaved functions of the solutions. Second, we integrate the problem over an interval that avoids the singularity. We are able to prove a uniform convergence result for the approximate solutions. We describe how the approximation is constructed for the various values of the deficiency index associated with the differential equation. The solution of the nonlinear problem is obtained by a globally convergent iterative method.     

B-spline collocation for solution of two-point boundary value problems

A numerical method based on B-spline is developed to solve the general nonlinear two-point boundary value problems up to order 6. The standard formulation of sextic spline for the solution of boundary value problems leads to non-optimal approximations. In order to derive higher orders of accuracy, high order perturbations of the problem are generated and applied to construct the numerical algorithm. The error analysis and convergence properties of the method are studied via Green’s function approach. O(h⁶) global error estimates are obtained for numerical solution of these classes of problems. Numerical results are given to illustrate the efficiency of the proposed method. Results of numerical experiments verify the theoretical behavior of the orders of convergence.     

Runge–Kutta methods and Banach algebras

The equations defining both the exact and the computed solution to an initial value problem are related to a single functional equation, which can be regarded as prototypical. The functional equation can be solved in terms of a formal Taylor series, which can also be generated using an iteration process. This leads to the formal Taylor expansions of the solution and approximate solutions to initial value problems. The usual formulation, using rooted trees, can be modified to allow for linear combinations of trees, and this gives an insight into the nature of order conditions for explicit Runge–Kutta methods. A short derivation of the family of fourth order methods with four stages is given.     

A family of multistep methods to integrate orbits on spheres

Summary A trajectory problem is an initial value problem where the interest lies in obtaining the curve traced by the solution, rather than in finding the actual correspondence between the values of the parameter and the points on that curve. This paper introduces a family of multi-stage, multi-step numerical methods to integrate trajectory problems whose solution is on a spherical surface. It has been shown that this kind of algorithms has good numerical properties: consistency, stability, convergence and others that are not standard. The latest ones make them a better choice for certain problems.     

Piecewise polynomial collocation for linear boundary value problems of fractional differential equations

We consider a class of boundary value problems for linear multi-term fractional differential equations which involve Caputo-type fractional derivatives. Using an integral equation reformulation of the boundary value problem, some regularity properties of the exact solution are derived. Based on these properties, the numerical solution of boundary value problems by piecewise polynomial collocation methods is discussed. In particular, we study the attainable order of convergence of proposed algorithms and show how the convergence rate depends on the choice of the grid and collocation points. Theoretical results are verified by two numerical examples.     

Weighted iterative solutions of linear differential equations and heat conduction of ideal gases in moment theory

An iterative method is proposed to find a particular solution of a system of linear differential equations, in the form of a fixed-point problem, with no boundary conditions. To circumvent the unboundedness of differential operators, iterative approximation with gradually decreasing weight is used. Conditions for convergence that can easily be checked in numerical iterations are established. Furthermore, for the numerical iterative scheme, uniqueness and stability theorems are proved. These results are applied to heat conduction of ideal gases in moment theory.     

A uniformly convergent difference scheme for a semilinear singular perturbation problem

Summary We present a difference scheme for solving a semilinear singular perturbation problem with any number of turning points of arbitrary orders. It is shown that a solution of the scheme converges, uniformly in a perturbation parameter, to that of the continuous problem.