A note on continuous dependence of solutions of dynamic equations on time scales

We give a precise formulation and a proof as constructive as possible of the widely accepted claim that solutions of a dynamic equation depend continuously on the base time scale. Our approach to this problem is via Euler polygons which opens possibilities for development of numerical analysis of dynamic equations on time scales.     

Runge-Kutta Methods Adapted to Manifolds and Based on Rigid Frames

We consider numerical integration methods for differentiable manifolds as proposed by Crouch and Grossman. The differential system is phrased by means of a system of frame vector fields E ₁, ... , E _n on the manifold. The numerical approximation is obtained by composing flows of certain vector fields in the linear span of E ₁, ... , E _n that are tangent to the differential system at various points. The methods reduce to traditional Runge-Kutta methods if the frame vector fields are chosen as the standard basis of euclidean ⁿ . A complete theory for the order conditions involving ordered rooted trees is developed. Examples of explicit and diagonal implicit methods are presented, along with some numerical results.     

Indexes and Special Discretization Methods for Linear partial Differential Algebraic Equations

Linear partial differential algebraic equations (PDAEs) of the form Au _t(t, x) + Bu _xx(t, x) + Cu(t, x) = f(t, x) are studied where at least one of the matrices A, B R ^n×n is singular. For these systems we introduce a uniform differential time index and a differential space index. We show that in contrast to problems with regular matrices A and B the initial conditions and/or boundary conditions for problems with singular matrices A and B have to fulfill certain consistency conditions. Furthermore, two numerical methods for solving PDAEs are considered. In two theorems it is shown that there is a strong dependence of the order of convergence on these indexes. We present examples for the calculation of the order of convergence and give results of numerical calculations for several aspects encountered in the numerical solution of PDAEs.     

Numerical Solution of the Bagley-Torvik Equation

We consider the numerical solution of the Bagley-Torvik equation Ay(t) + BD _* ^3/2 y(t) + Cy(t) = f(t), as a prototype fractional differential equation with two derivatives. Approximate solutions have recently been proposed in the book and papers of Podlubny in which the solution obtained with approximate methods is compared to the exact solution. In this paper we consider the reformulation of the Bagley-Torvik equation as a system of fractional differential equations of order 1/2. This allows us to propose numerical methods for its solution which are consistent and stable and have arbitrarily high order. In this context we specifically look at fractional linear multistep methods and a predictor-corrector method of Adams type.     

A new stable,explicit, and generic third-order method for simulating conductive heat transfer

In this paper we introduce a new type of explicit numerical algorithm to solve the spatially discretized linear heat or diffusion equation. After discretizing the space variables as in standard finite difference methods, this novel method does not approximate the time derivatives by finite differences, but use three stage constant-neighbor and linear neighbor approximations to decouple the ordinary differential equations and solve them analytically. In the final expression for the new values of the variable, the time step size appears not in polynomial or rational, but in exponential form with negative coefficients, which can guarantee unconditional stability. The scheme contains a free parameter p. We show that the convergence of the method is third-order in the time step size regardless of the values of p, and, according to von Neumann stability analysis, the method is stable for a wide range of p. We validate the new method by testing the results in a case where the analytical solution exists, then we demonstrate the competitiveness by comparing its performance with several other numerical solvers.     

The numerical solution of fractional differential equations: Speed versus accuracy

This paper is concerned with the development of efficient algorithms for the approximate solution of fractional differential equations of the form D y(t)=f(t,y(t)), R ⁺–N.()We briefly review standard numerical techniques for the solution of () and we consider how the computational cost may be reduced by taking into account the structure of the calculations to be undertaken. We analyse the fixed memory principle and present an alternative nested mesh variant that gives a good approximation to the true solution at reasonable computational cost. We conclude with some numerical examples.     

A structure‐preserving doubling algorithm for Lur'e equations

We introduce a numerical method for the numerical solution of the Lur'e equations, a system of matrix equations that arises, for instance, in linear‐quadratic infinite time horizon optimal control. We focus on small‐scale, dense problems. Via a Cayley transformation, the problem is transformed to the discrete‐time case, and the structural infinite eigenvalues of the associated matrix pencil are deflated. The deflated problem is associated with a symplectic pencil with several Jordan blocks of eigenvalue 1 and even size, which arise from the nontrivial Kronecker chains at infinity of the original problem. For the solution of this modified problem, we use the structure‐preserving doubling algorithm. Implementation issues such as the choice of the parameter γ in the Cayley transform are discussed. The most interesting feature of this method, with respect to the competing approaches, is the absence of arbitrary rank decisions, which may be ill‐posed and numerically troublesome. The numerical examples presented confirm the effectiveness of this method. Copyright © 2015 John Wiley & Sons, Ltd.     

Superconvergence in Collocation Methods on Quasi-Graded Meshes for Functional Differential Equations with Vanishing Delays

We study the optimal orders of (global and local) superconvergence in piecewise polynomial collocation on quasi-graded meshes for functional differential equations with (nonlinear) delays vanishing at t=0. It is shown that while for linear delays (e.g. proportional delays qt with 0<q<1) and certain nonlinear delays the classical optimal order results still hold, high degree of tangency with the identity function at t=0 leads not only to a reduction in the order of superconvergence but also to very serious difficulties in the actual computation of numerical approximations. AMS subject classification (2000) 65R20, 34K28     

Quintic spline technique for time fractional fourth‐order partial differential equation

Higher order non‐Fickian diffusion theories involve fourth‐order linear partial differential equations and their solutions. A quintic polynomial spline technique is used for the numerical solutions of fourth‐order partial differential equations with Caputo time fractional derivative on a finite domain. These equations occur in many applications in real life problems such as modeling of plates and thin beams, strain gradient elasticity, and phase separation in binary mixtures, which are basic elements in engineering structures and are of great practical significance to civil, mechanical, and aerospace engineering. The quintic polynomial spline technique is used for space discretization and the time‐stepping is done using a backward Euler method based on the L1 approximation to the Caputo derivative. The stability and convergence analysis are also discussed. The numerical results are given, which demonstrate the effectiveness and accuracy of the numerical method. The numerical results obtained in this article are also compared favorably well with the results of (S. S. Siddiqi and S. Arshed, Int. J. Comput. Math. 92 (2015), 1496–1518). © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 445–466, 2017     

A Dimensional Splitting Method for Quasilinear Hyperbolic Equations with Variable Coefficients

A numerical method is presented for the variable coefficient, nonlinear hyperbolic equation u _t + _i=1 ^d V _i(x, t)f _i(u) _{x i} = 0 in arbitrary space dimension for bounded velocities that are Lipschitz continuous in the x variable. The method is based on dimensional splitting and uses a recent front tracking method to solve the resulting one-dimensional non-conservative equations. The method is unconditionally stable, and it produces a subsequence that converges to the entropy solution as the discretization of time and space tends to zero. Four numerical examples are presented; numerical error mechanisms are illustrated for two linear equations, the efficiency of the method compared with a high-resolution TVD method is discussed for a nonlinear problem, and finally, applications to reservoir simulation are presented.     

Multi-Step Maruyama Methods for Stochastic Delay Differential Equations

Abstract

In this article the numerical approximation of solutions of Itô stochastic delay differential equations is considered. We construct stochastic linear multi-step Maruyama methods and develop the fundamental numerical analysis concerning their 𝕃 _p -consistency, numerical 𝕃 _p -stability and 𝕃 _p -convergence. For the special case of two-step Maruyama schemes we derive conditions guaranteeing their mean-square consistency.     

On the numerical solution of stochastic quadratic integral equations via operational matrix method

In this paper, stochastic operational matrix of integration based on delta functions is applied to obtain the numerical solution of linear and nonlinear stochastic quadratic integral equations (SQIEs) that appear in modelling of many real problems. An important advantage of this method is that it dose not need any integration to compute the constant coefficients. Also, this method can be utilized to solve both linear and nonlinear problems. By using stochastic operational matrix of integration together collocation points, solving linear and nonlinear SQIEs converts to solve a nonlinear system of algebraic equations, which can be solved by using Newton's numerical method. Moreover, the error analysis is established by using some theorems. Also, it is proved that the rate of convergence of the suggested method is O(h²). Finally, this method is applied to solve some illustrative examples including linear and nonlinear SQIEs. Numerical experiments confirm the good accuracy and efficiency of the proposed method.     

A new two-point scheme for multiple roots of nonlinear equations

In most of the earlier research for multiple zeros, in order to obtain a new iteration function from the existing scheme, the usual practice is to make no change at the first substep. In this paper, we explore the idea that what are the advantages if the flexibility of choice is also given at the first substep. Therefore, we present a new two-point sixth-order scheme for multiple roots (m>1). The main advantages of our scheme over the existing schemes are flexibility at both substeps, simple body structure, smaller residual error, smaller error difference between two consecutive iterations, and smaller asymptotic error constant. The development of the scheme is based on midpoint formula and weight functions of two variables. We compare our methods with the existing methods of the same order with real-life applications as well as standard test problems. From the numerical results, we find that our methods can be considered as better alternates for the existing methods of the same order. Finally, dynamical study of the proposed schemes is presented that confirms the theoretical results.     

A fully discrete H 1-Galerkin method with quadrature for nonlinear advection–diffusion–reaction equations

We propose and analyze a fully discrete H ¹-Galerkin method with quadrature for nonlinear parabolic advection–diffusion–reaction equations that requires only linear algebraic solvers. Our scheme applied to the special case heat equation is a fully discrete quadrature version of the least-squares method. We prove second order convergence in time and optimal H ¹ convergence in space for the computer implementable method. The results of numerical computations demonstrate optimal order convergence of scheme in H ^k for k = 0, 1, 2. Support of the Australian Research Council is gratefully acknowledged.     

Strongly nonlinear modal equations for nearly integrable PDEs

Summary The purpose of this paper is the derivation of reduced, finite-dimensional dynamical systems that govern the near-integrable modulations ofN-phase, spatially periodic, integrable wavetrains. The small parameter in this perturbation theory is the size of the nonintegrable perturbation in the equation, rather than the amplitude of the solution, which is arbitrary. Therefore, these reduced equations locally approximate strongly nonlinear behavior of the nearly integrable PDE. The derivation we present relies heavily on the integrability of the underlying PDE and applies, in general, to anyN-phase periodic wavetrain. For specific applications, however, a numerical pretest is applied to fix the truncation orderN. We present one example of the reduction philosophy with the damped, driven sine-Gordon system and summarize our present progress toward application of the modulation equations to this numerical study.     

Using successive approximations for improving the convergence of GMRES method

In this paper, our attention is concentrated on the GMRES method for the solution of the system (I–T)x=b of linear algebraic equations with a nonsymmetric matrix. We perform m pre-iterations y _l+1 =T _yl +b before starting GMRES and put y _m for the initial approximation in GMRES. We derive an upper estimate for the norm of the error vector in dependence on the mth powers of eigenvalues of the matrix T Further we study under what eigenvalues lay-out this upper estimate is the best one. The estimate shows and numerical experiments verify that it is advisable to perform pre-iterations before starting GMRES as they require fewer arithmetic operations than GMRES. Towards the end of the paper we present a numerical experiment for a system obtained by the finite difference approximation of convection-diffusion equations.     

Stability Analysis of Runge-Kutta Methods for Non-Linear Delay Differential Equations

This paper is concerned with the numerical solution of delay differential equations(DDEs). We focus on the stability behaviour of Runge-Kutta methods for nonlinear DDEs. The new concepts of GR(l)-stability, GAR(l)-stability and weak GAR(l)-stability are further introduced. We investigate these stability properties for (k, l)-algebraically stable Runge-Kutta methods with a piecewise constant or linear interpolation procedure.     

An exact Riemann solver and a Godunov scheme for simulating highly transient mixed flows

The current research aims at deriving a one-dimensional numerical model for describing highly transient mixed flows. In particular, this paper focuses on the development and assessment of a unified numerical scheme adapted to describe free-surface flow, pressurized flow and mixed flow (characterized by the simultaneous occurrence of free-surface and pressurized flows). The methodology includes three steps. First, the authors derived a unified mathematical model based on the Preissmann slot model. Second, a first-order explicit finite volume Godunov-type scheme is used to solve the set of equations. Third, the numerical model is assessed by comparison with analytical, experimental and numerical results. The key results of the paper are the development of an original negative Preissmann slot for simulating sub-atmospheric pressurized flow and the derivation of an exact Riemann solver for the Saint-Venant equations coupled with the Preissmann slot.     

On the accuracy of the viscous form in simulations of incompressible flow problems

We present a numerical study of drag/lift and flux estimates using two forms of Navier‐Stokes equations (NSE) that are equivalent in the continuum formulation but not in the discrete finite element formulation. The two investigated forms of the NSE differ in the viscous term that is represented in one form by νΔ u with ν being the viscosity and 2ν?·?^S u in the other form where ?^S represents the deformation tensor. The study consists of numerical analysis of the two forms and computations of drag/lift, pressure drop on the cylinder problem and computations of flux for the Poiseuille flow. The main objective is to provide a clear comparison of the reference values for the maximal drag and lift coefficient at the cylinder and for the pressure difference between the front and the back of the cylinder at the final time for the two forms of NSEs. Our computational results of the reference values do not differ significantly between the two forms, but the differences are there. For the Poiseuille flow, the differences in the flux computations were much smaller, and this agreed with the computationally obtained results of the divergence of the velocity field. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 523–541, 2012     

Numerical method for coupling the macro and meso scales in stochastic chemical kinetics

A numerical method is developed for simulation of stochastic chemical reactions. The system is modeled by the Fokker–Planck equation for the probability density of the molecular state. The dimension of the domain of the equation is reduced by assuming that most of the molecular species have a normal distribution with a small variance. The numerical approximation preserves properties of the analytical solution such as non-negativity and constant total probability. The method is applied to a nine dimensional problem modelling an oscillating molecular clock. The oscillations stop at a fixed point with a macroscopic model but they continue with our two dimensional, mixed macroscopic and mesoscopic model. Dedicated to the memory of Germund Dahlquist (1925–2005). AMS subject classification (2000)  65M20, 65M60