Hybrid,phase–fitted,four–step methods of seventh order for solving x″(t) = f(t,x)

A four‐step method of seventh algebraic order is presented. It is tuned for addressing the special second order initial value problem. The new method is hybrid, explicit, and uses three stages per step. In addition is phase fitted. In consequence it uses variable coefficients that depend on the magnitude of the step‐size. We also present numerical tests on a set of standard problems that illustrate the efficiency of the derived method over older ones given in the relevant literature.     

Step size control in the numerical solution of stochastic differential equations

We introduce a variable step size algorithm for the pathwise numerical approximation of solutions to stochastic ordinary differential equations. The algorithm is based on a new pair of embedded explicit Runge-Kutta methods of strong order 1.5(1.0), where the method of strong order 1.5 advances the numerical computation and the difference between approximations defined by the two methods is used for control of the local error. We show that convergence of our method is preserved though the discretization times are not stopping times any more, and further, we present numerical results which demonstrate the effectiveness of the variable step size implementation compared to a fixed step size implementation.     

A second‐order ensemble method based on a blended backward differentiation formula timestepping scheme for time‐dependent Navier–Stokes equations

We present a second‐order ensemble method based on a blended three‐step backward differentiation formula (BDF) timestepping scheme to compute an ensemble of Navier–Stokes equations. Compared with the only existing second‐order ensemble method that combines the two‐step BDF timestepping scheme and a special explicit second‐order Adams–Bashforth treatment of the advection term, this method is more accurate with nominal increase in computational cost. We give comprehensive stability and error analysis for the method. Numerical examples are also provided to verify theoretical results and demonstrate the improved accuracy of the method. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 34–61, 2017     

A tau method for nonlinear dynamical systems

In this work we present a new Tau method for the solution of nonlinear systems of differential equations which are linear in the derivative of highest order and polynomial in the remaining. We avoid the linearization of the problem by associating to it a nonlinear algebraic system and combine a forward substitution with the Tau method. We develop an adaptive step by step version of this alternative nonlinear tau method and we apply it to several nonlinear dynamical systems.     

Optimized higher order time discretization of second order hyperbolic problems: Construction and numerical study

We investigate explicit higher order time discretizations of linear second order hyperbolic problems. We study the even order (2m) schemes obtained by the modified equation method. We show that the corresponding CFL upper bound for the time step remains bounded when the order of the scheme increases. We propose variants of these schemes constructed to optimize the CFL condition. The corresponding optimization problem is analyzed in detail. The optimal schemes are validated through various numerical results.     

General linear methods: connection to one step methods and invariant curves

Summary We generalize a result of Kirchgraber (1986) on multistep methods. We show that every strictly stable general linear method is essentially conjugate to a one step method of the same order. This result may be used to show that general properties of one step methods carry over to general linear methods. As examples we treat the existence of invariant curves and the construction of attracting sets.     

On the Convergence Analysis of the Inexact Linearly Implicit Euler Scheme for a Class of Stochastic Partial Differential Equations

This paper is concerned with the adaptive numerical treatment of stochastic partial differential equations. Our method of choice is Rothe’s method. We use the implicit Euler scheme for the time discretization. Consequently, in each step, an elliptic equation with random right-hand side has to be solved. In practice, this cannot be performed exactly, so that efficient numerical methods are needed. Well-established adaptive wavelet or finite-element schemes, which are guaranteed to converge with optimal order, suggest themselves. We investigate how the errors corresponding to the adaptive spatial discretization propagate in time, and we show how in each time step the tolerances have to be chosen such that the resulting perturbed discretization scheme realizes the same order of convergence as the one with exact evaluations of the elliptic subproblems.     

LOCAL PRESSURE CORRECTION FOR THE STOKES SYSTEM

Pressure correction methods constitute the most widely used solvers for the timedependent Navier-Stokes equations.There are several different pressure correction methods,where each time step usually consists in a predictor step for a non-divergence-free velocity,followed by a Poisson problem for the pressure(or pressure update),and a final velocity correction to obtain a divergence-free vector field.In some situations,the equations for the velocities are solved explicitly,so that the numerical most expensive step is the elliptic pressure problem.We here propose to solve this Poisson problem by a domain decomposition method which does not need any communication between the sub-regions.Hence,this system is perfectly adapted for parallel computation.We show under certain assumptions that this new scheme has the same order of convergence as the original pressure correction scheme(with global projection).Numerical examples for the Stokes system show the effectivity of this new pressure correction method.The convergence order O(k^2)for resulting velocity fields can be observed in the norm l^2(0,T;L^2(Ω)).     

Implicit peer methods for large stiff ODE systems

Implicit two-step peer methods are introduced for the solution of large stiff systems. Although these methods compute s-stage approximations in each time step one-by-one like diagonally-implicit Runge-Kutta methods the order of all stages is the same due to the two-step structure. The nonlinear stage equations are solved by an inexact Newton method using the Krylov solver FOM (Arnoldi??s method). The methods are zero-stable for arbitrary step size sequences. We construct different methods having order p=s in the multi-implicit case and order p=s?1 in the singly-implicit case with arbitrary step sizes and s??5. Numerical tests in Matlab for several semi-discretized partial differential equations show the efficiency of the methods compared to other Krylov codes.     

Finite Element Methods for Parabolic Stochastic PDE’s

We study the rate of convergence of some explicit and implicit numerical schemes for the solution of a parabolic stochastic partial differential equation driven by white noise. These include the forward and backward Euler and the Crank–Nicholson schemes. We use the finite element method. We find, as expected, that the rates of convergence are substantially similar to those found for finite difference schemes, at least when the size of the time step k is on the order of the square of the size of the space step h: all the schemes considered converge at a rate on the order of h^1/2+k^1/4, which is known to be optimal. We also consider cases where k is much greater than h², and find that only the backward Euler method always attains the optimal rate; other schemes, even though they are stable, can fail to convergence to the true solution if the time step is too long relative to the space step. The Crank–Nicholson scheme behaves particularly badly in this case, even though it is a higher-order method. Mathematics Subject Classifications (2000) 60H15, 60H35, 65N30, 35R60.     

MIXED FINITE ELEMENT METHODS FOR FRACTIONAL NAVIER-STOKES EQUATIONS

This paper gives the detailed numerical analysis of mixed finite element method for fractional Navier-Stokes equations.The proposed method is based on the mixed finite element method in space and a finite difference scheme in time.The stability analyses of semi-discretization scheme and fully discrete scheme are discussed in detail.Furthermore,We give the convergence analysis for both semidiscrete and flly discrete schemes and then prove that the numerical solution converges the exact one with order O(h2+k),where h and k:respectively denote the space step size and the time step size.Finally,numerical examples are presented to demonstrate the effectiveness of our numerical methods.     

A method for the localization of singularities of a solution to a convolution-type equation of the first kind with a step kernel

We consider the problem of the localization of singularities (delta-functions) of a solution to a convolution-type equation of the first kind with a step kernel. We propose a regularization method which allows one to calculate the number of singularities, to approximate their location, and to estimate the approximation error. We also adduce bounds for an important characteristic of the method, namely, the separability threshold. We prove the order optimality of the proposed method on classes of functions with singularities both with respect to the accuracy and the separability.     

Block s‐step Krylov iterative methods

Block (including s‐step) iterative methods for (non)symmetric linear systems have been studied and implemented in the past. In this article we present a (combined) block s‐step Krylov iterative method for nonsymmetric linear systems. We then consider the problem of applying any block iterative method to solve a linear system with one right‐hand side using many linearly independent initial residual vectors. We present a new algorithm which combines the many solutions obtained (by any block iterative method) into a single solution to the linear system. This approach of using block methods in order to increase the parallelism of Krylov methods is very useful in parallel systems. We implemented the new method on a parallel computer and we ran tests to validate the accuracy and the performance of the proposed methods. It is expected that the block s‐step methods performance will scale well on other parallel systems because of their efficient use of memory hierarchies and their reduction of the number of global communication operations over the standard methods. Copyright © 2009 John Wiley & Sons, Ltd.     

Piecewise finite series solution of nonlinear initial value differential problem

In this paper, we apply a piecewise finite series as a hybrid analytical-numerical technique for solving some nonlinear systems of ordinary differential equations. The finite series is generated by using the Adomian decomposition method, which is an analytical method that gives the solution based on a power series and has been successfully used in a wide range of problems in applied mathematics. We study the influence of the step size and the truncation order of the piecewise finite series Adomian (PFSA) method on the accuracy of the solutions when applied to nonlinear ODEs. Numerical comparisons between the PFSA method with different time steps and truncation orders against Runge-Kutta type methods are presented. Based on the numerical results we propose a low value truncation order approach with small time step size. The numerical results show that the PFSA method is accurate and easy to implement with the proposed approach.     

Half-explicit Runge-Kutta methods with explicit stages for differential-algebraic systems of index 2

Usually the straightforward generalization of explicit Runge-Kutta methods for ordinary differential equations to half-explicit methods for differential-algebraic systems of index 2 results in methods of orderq≤2. The construction of higher order methods is simplified substantially by a slight modification of the method combined with an improved strategy for the computation of the algebraic solution components. We give order conditions up to orderq=5 and study the convergence of these methods. Based on the fifth order method of Dormand and Prince the fifth order half-explicit Runge-Kutta method HEDOP5 is constructed that requires the solution of 6 systems of nonlinear equations per step of integration.     

Finite element method for Grwünwald–Letnikov time-fractional partial differential equation

In this article, we consider the finite element methods (FEM) for Grwünwald–Letnikov time-fractional diffusion equation, which is obtained from the standard two-dimensional diffusion equation by replacing the first-order time derivative with a fractional derivative (of order α, with 0?h ^r+1?+?τ^2-α), where h, τ and r are the space step size, time step size and polynomial degree, respectively. A numerical example is presented to verify the order of convergence.     

A nonmonotone truncated Newton–Krylov method exploiting negative curvature directions, for large scale unconstrained optimization

We propose a new truncated Newton method for large scale unconstrained optimization, where a Conjugate Gradient (CG)-based technique is adopted to solve Newton’s equation. In the current iteration, the Krylov method computes a pair of search directions: the first approximates the Newton step of the quadratic convex model, while the second is a suitable negative curvature direction. A test based on the quadratic model of the objective function is used to select the most promising between the two search directions. Both the latter selection rule and the CG stopping criterion for approximately solving Newton’s equation, strongly rely on conjugacy conditions. An appropriate linesearch technique is adopted for each search direction: a nonmonotone stabilization is used with the approximate Newton step, while an Armijo type linesearch is used for the negative curvature direction. The proposed algorithm is both globally and superlinearly convergent to stationary points satisfying second order necessary conditions. We carry out a significant numerical experience in order to test our proposal.     

Evaluation of the local error of fractional-rational multistep methods with variable integration step

To ensure the proper qualitative characteristic of approximate numerical solution of the Cauchy problem for a system of ordinary differential equations, it is necessary to formulate certain conditions that have to be satisfied by numerical methods. The efficiency of a numerical method is determined by constructing the algorithm of integration step changing and the choice of the order of the method. The construction of such a method requires one to determine preliminarily the admissible error of the method in each integration step. A theorem on the evaluation of the local error of multistep numerical p th-order methods with variable integration step without taking into account the round-off error is formulated. This theorem enables one to construct an efficient algorithm for the step change and the choice of the corresponding order of the method.     

Improving the accuracy of BDF methods for index 3 differential-algebraic equations

Methods for solving index 3 DAEs based on BDFs suffer a loss of accuracy when there is a change of step size or a change of order of the method. A layer of nonuniform convergence is observed in these cases, andO(1) errors may appear in the algebraic variables. From the viewpoint of error control, it is beneficial to allow smooth changes of step size, and since most codes based on BDFs are of variable order, it is also of interest to avoid the inaccuracies caused by a change of order of the method. In the case of BDFs applied to index 3 DAEs in semi-explicit form, we present algorithms that correct toO(h) the inaccurate approximations to the algebraic variables when there are changes of step size in the backward Euler method. These algorithms can be included in an existing code at a very small cost. We have also described how to obtain formulas that correct theO(1) errors in the algebraic variables appearing after a change of order.This author thanks the Centro de Estadística y Software Matemático de la Universidad Simón Bolívar (CESMa) for permitting her free use of its research facilities.     

On the exactness of the ε-constraint method for biobjective nonlinear integer programming

The ε-constraint method is a well-known scalarization technique used for multiobjective optimization. We explore how to properly define the step size parameter of the method in order to guarantee its exactness when dealing with biobjective nonlinear integer problems. Under specific assumptions, we prove that the number of subproblems that the method needs to address to detect the complete Pareto front is finite. We report numerical results on portfolio optimization instances built on real-world data and show a comparison with an existing criterion space algorithm.