The time discretization in classes of integro‐differential equations with completely monotonic kernels: Weighted asymptotic convergence

We consider the time discretization for the solution of the equation with . Here, the operators L_j are densely defined positive self‐adjoint linear operator on a Hilbert space H and have spectral decompositions with respect to a common resolution of the identity in H . The kernel functions , are assumed to be completely monotonic on (0,∞) and locally integrable, but not constant. The considered time discretization method comes from [Da Xu, Science China Mathematics 56 (2013), 395–424], where the backward Euler method is combined with order one convolution quadrature for approximating the integral term. In this article, the convergence properties of the time discretization are given in the weighted and norm, where ρ is a given weighted function. Numerical experiments show the theoretical results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 896–935, 2016     

Low-storage Runge–Kutta methods for stochastic differential equations

Runge–Kutta methods that require only two memory locations per variable and have strong local order γ=1.5 for non-commutative systems of stochastic differential equations driven by one Wiener process are devised in this paper. A first step in the derivation is to extend existing deterministic methods to the commutative stochastic case, for which higher accuracy is also obtained. Numerical results are presented to validate the approach.     

Runge–Kutta characteristic methods for first-order linear hyperbolic equations

We develop two Runge–Kutta characteristic methods for the solution of the initial-boundary value problems for first-order linear hyperbolic equations. One of the methods is based on a backtracking of the characteristics, while the other is based on forward tracking. The derived schemes naturally incorporate inflow boundary conditions into their formulations and do not need any artificial outflow boundary condition. They are fully mass conservative and can be viewed as higher-order time integration schemes improved over the ELLAM (Eulerian–Lagrangian localized adjoint method) method developed previously. Moreover, they have regularly structured, well-conditioned, symmetric, and positive-definite coefficient matrices. Extensive numerical results are presented to compare the performance of these methods with many well studied and widely used methods, including the Petrov–Galerkin methods, the streamline diffusion methods, the continuous and discontinuous Galerkin methods, the MUSCL, and the ENO schemes. The numerical experiments also verify the optimal-order convergence rates of the Runge–Kutta methods developed in this article. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 617–661, 1997     

Dissipativity of Runge–Kutta methods for neutral delay integro-differential equations

This paper is concerned with the numerical dissipativity of a class of nonlinear neutral delay integro-differential equations. The dissipativity results are obtained for algebraically stable Runge–Kutta methods when they are applied to above problems.     

Numerical solution of Volterra integral and integro-differential equations with rapidly vanishing convolution kernels

Variable stepsize algorithms for the numerical solution of nonlinear Volterra integral and integro-differential equations of convolution type are described. These algorithms are based on an embedded pair of Runge–Kutta methods of order p=5 and p=4 proposed by Dormand and Prince with interpolation of uniform order p=4. They require O(N) number of kernel evaluations, where N is the number of steps. The cost of the algorithms can be further reduced for equations that have rapidly vanishing convolution kernels, by using waveform relaxation iterations after computing the numerical approximation by variable stepsize algorithm on some initial interval. AMS subject classification (2000)  65R20, 45L10, 93C22     

Fast Runge–Kutta methods for nonlinear convolution systems of Volterra integral equations

In this paper fast implicit and explicit Runge–Kutta methods for systems of Volterra integral equations of Hammerstein type are constructed. The coefficients of the methods are expressed in terms of the values of the Laplace transform of the kernel. These methods have been suitably constructed in order to be implemented in an efficient way, thus leading to a very low computational cost both in time and in space. The order of convergence of the constructed methods is studied. The numerical experiments confirm the expected accuracy and computational cost. AMS subject classification (2000)  65R20, 45D05, 44A35, 44A10     

Uniform l1 behaviour for time discretization of a Volterra equation with completely monotonic kernel: I. stability

This paper is the first of two papers on the time discretizationof the equation u_t + ^t ₀ ß (t — s) Au (s) ds= 0, t > 0, u (0) = u₀, where A is a self-adjoint denselydefined linear operator on a Hilbert space H with a completeeigensystem {_m, _m}_{m = 1}, and ß (t) is completely monotonicand locally integrable, but not constant. The equation is discretizedin time using first-order differences in combination with order-oneconvolution quadrature. The stability properties of the timediscretization are derived in the l¹_t (0, ; H) norm.     

Runge–Kutta methods for first-order periodic boundary value differential equations with piecewise constant arguments

In this paper we deal with the numerical solutions of Runge–Kutta methods for first-order periodic boundary value differential equations with piecewise constant arguments. The numerical solution is given by the numerical Green’s function. It is shown that Runge–Kutta methods preserve their original order for first-order periodic boundary value differential equations with piecewise constant arguments. We give the conditions under which the numerical solutions preserve some properties of the analytic solutions, e.g., uniqueness and comparison theorems. Finally, some experiments are given to illustrate our results.     

Generalized Kronecker product splitting iteration for the solution of implicit Runge–Kutta and boundary value methods

This paper is concerned with a generalization of the Kronecker product splitting (KPS) iteration for solving linear systems arising in implicit Runge–Kutta and boundary value methods discretizations of ordinary differential equations. It is shown that the new scheme can outperform the standard KPS method in some situations and can be used as an effective preconditioner for Krylov subspace methods. Numerical experiments are presented to demonstrate the effectiveness of the methods. Copyright © 2014 John Wiley & Sons, Ltd.     

The boundaries of the solutions of the linear Volterra integral equations with convolution kernel

Some boundaries about the solution of the linear Volterra integral equations of the second type with unit source term and positive monotonically increasing convolution kernel were obtained in Ling, 1978 and 1982. A method enabling the expansion of the boundary of the solution function of an equation in this type was developed in I. Özdemir and Ö. F. Temizer, 2002.

In this paper, by using the method in Özdemir and Temizer, it is shown that the boundary of the solution function of an equation in the same form can also be expanded under different conditions than those that they used.

     

Construction of highly stable parallel two-step Runge–Kutta methods for delay differential equations

It is shown that any A-stable two-step Runge–Kutta method of order and stage order for ordinary differential equations can be extended to the P-stable method of uniform order for delay differential equations.     

Numerical treatment of retarded boundary integral equations by sparse panel clustering

Stefan Sauter Institute for Mathematics, University of Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland ^{We consider the wave equation in a boundary integral formulation.The discretization in time is done by using convolution quadraturetechniques and a Galerkin boundary element method for the spatialdiscretization. In a previous paper, we have introduced a sparseapproximation of the system matrix by cut-off, in order to reducethe storage costs. In this paper, we extend this approach byintroducing a panel clustering method to further reduce thesecosts.}     

Convolution quadrature time discretization of fractional diffusion-wave equations

We propose and study a numerical method for time discretization of linear and semilinear integro-partial differential equations that are intermediate between diffusion and wave equations, or are subdiffusive. The method uses convolution quadrature based on the second-order backward differentiation formula. Second-order error bounds of the time discretization and regularity estimates for the solution are shown in a unified way under weak assumptions on the data in a Banach space framework. Numerical experiments illustrate the theoretical results.

     

Numerical solution of Volterra integral equations with singularities

The numerical solution of linear Volterra integral equations of the second kind is discussed. The kernel of the integral equation may have weak diagonal and boundary singularities. Using suitable smoothing techniques and polynomial splines on mildly graded or uniform grids, the convergence behavior of the proposed algorithms is studied and a collection of numerical results is given.     

Numerical solution of Hammerstein integral equations by using combination of spline-collocation method and Lagrange interpolation

The numerical solutions to the nonlinear integral equations of the Hammerstein-type:

with using B-splines functions are investigated. An interpolation method based on B-splines functions combined with a new collocation method is presented. Finally, for showing efficiency of the method we give some numerical examples.     

Numerical solution of linear Volterra integral equations of the second kind with sharp gradients

Collocation methods are a well-developed approach for the numerical solution of smooth and weakly singular Volterra integral equations. In this paper, we extend these methods through the use of partitioned quadrature based on the qualocation framework, to allow the efficient numerical solution of linear, scalar Volterra integral equations of the second kind with smooth kernels containing sharp gradients. In this case, the standard collocation methods may lose computational efficiency despite the smoothness of the kernel. We illustrate how the qualocation framework can allow one to focus computational effort where necessary through improved quadrature approximations, while keeping the solution approximation fixed. The computational performance improvement introduced by our new method is examined through several test examples. The final example we consider is the original problem that motivated this work: the problem of calculating the probability density associated with a continuous-time random walk in three dimensions that may be killed at a fixed lattice site. To demonstrate how separating the solution approximation from quadrature approximation may improve computational performance, we also compare our new method to several existing Gregory, Sinc, and global spectral methods, where quadrature approximation and solution approximation are coupled.     

On some properties of systems of Volterra integral equations of the fourth kind with kernel of convolution type

We consider the system of integral equations of the form Ax +V x = Ψ, where V is the Volterra operator with kernel of convolution type and A is a constant matrix, det A = 0. We prove an existence theorem and establish necessary and sufficient conditions for the kernel of the operator of the system to be trivial.     

Numerical solution of modified Black–Scholes equation pricing stock options with discrete dividend

This paper deals with the numerical solution of the modified Black–Scholes equation modelling the valuation of stock options with discrete dividend payments. By using a delta-defining sequence of the involved generalized Dirac delta function and applying the Mellin transform, an integral formula for the solution is obtained. Then, numerical quadrature approximations and illustrative examples are given.     

Numerical solution of Volterra integral and integro-differential equations of convolution type by using operational matrices of piecewise constant orthogonal functions

In this paper, we use operational matrices of piecewise constant orthogonal functions on the interval [0,1)$[0,1)$ to solve Volterra integral and integro-differential equations of convolution type without solving any system. We first obtain Laplace transform of the problem and then we find numerical inversion of Laplace transform by operational matrices. Numerical examples show that the approximate solutions have a good degree of accuracy.     

A direct method of solving singular integral equations with the Hilbert kernel

In this paper we consider a class of second-kind singular integral equations with Hilbert kernel on the unit circumference. We theoretically substantiate a solution method based on an interpolation-type operator.