Lod generalized trapezoidal formula schemes for parabolic differential equations in two space dimensions

We describe locally one-dimensional (LOD) time integration schemes for parabolic differential equations in two space dimensions, based on the generalized trapezoidal formulas (GTF(α)). We describe the schemes for the diffusion equation with Dirichlet and Neumann boundary conditions, for nonlinear reaction-diffusion equations, and for the convection-diffusion equation in two space dimensions. The obtained schemes are second order in time and unconditionally stable for all α ∈ [0, 1]. Numerical experiments are given to illustrate the obtained schemes and to compare their performance with the better known LOD Crank-Nicolson scheme. While the LOD Crank-Nicolson scheme can give unwanted oscillations in the computed solution, our present LOD-GTF(α) schemes provide both stable and accurate approximations for the true solution.     

Euler-like discrete models of the logistic differential equation

To understand the behavior of difference schemes on nonlinear differential equations, it seems desirable to extend the standard linear stability theory into a nonlinear theory. As a step in that direction, we investigate the stability properties of Euler-related integration algorithms by checking how they preserve and violate the dynamical structure of the logistic differential equation.Among the schemes considered are two linearly implicit nonstandard schemes which are adjoint to each other. We find that these schemes are superior to explicit schemes when they are stable and the blow-up time has not passed: for these λh-values they are dynamically faithful. When these schemes ‘turn unstable’, however, they have much less desirable properties than explicit or fully implicit schemes: they become simultaneously superstable and unstable. This is explained by the fact that these schemes are not self-adjoint: the linearly implicit self-adjoint scheme is dynamically faithful in an Euler-typical range of step sizes and gives correct stability for all step sizes.     

A class of generalized trapezoidal formulas for the numerical integration of

The classical arithmetic mean (AM) trapezoidal formula is known to be A-stable. Recently, Chawla and Al-Zanaidi [1] described a modified arithmetic mean (MAM) trapezoidal formula which is L-stable. In the present paper we introduce a one-parameter family of generalized trapezoidal formulas (GTFs), which include both the AM and MAM trapezoidal formulas as special cases. A GTF can switch from an A-stable to an L-stable method depending on parameter values, and it is shown to perform, for certain selections of the parameter values, better than AM and MAM formulas for the integration of examples of the stiff systems considered.     

Generalized trapezoidal formulas for parabolic equations

We investigate the application of the one-parameter family of generalized trapezoidal formulas (GTFs) introduced in Chawla et al. [2] for the time-integration of parabolic equations. The resulting GTF finite-difference schemes (GTF-FDS) are, in general, second order in both time and space and unconditionally stable. Interestingly, there exists a method of the family which is third order in time. Unlike the popular Crank -Nicolson scheme, our present GTF-FDS can cope with discontinuities in the boundary conditions and the initial conditions. We consider extensions of the GTF-FDS for equations with derivative boundary conditions and to a nonlinear problem. Numerical experiments demonstrate the superiority of the present GTF-FDS, especially for the case of problems with discontinuities in the boundary and the initial conditions.     

Tridiagonal solver-based L-stable one-step schemes for nonlinear parabolic equations

An attractive feature of the widely used Crank-Nicolson (C-N) scheme for parabolic equations is that it is a tridiagonal solver-based (TSB) scheme. But, in case of inconsistencies in the initial and boundary conditions or when the ratio of temporal to spatial steplengths is large, it can produce unwanted oscillations or an unacceptable solution. As alternative to C-N, Chawla et al. [2, 3] introduced L-stable generalized trapezoidal formulas (GTF(α)) which can give a more acceptable solution by a judicious choice of the parameter α; however, GTF are not TSB schemes. It is natural to ask for L-stable TSB schemes. In the present paper, we first introduce a one-parameter family of generalized midpoint formulas (GMF(α)); again GMF are not TSB schemes. We then introduce a two-parameter family through a linear combination of the GMF and the classical trapezoidal formula, and show the existence of a one-parameter subfamily of L-stable TSB schemes; these schemes are unconditionally stable. The computational performance of the obtained schemes is compared with the C-N scheme by considering a nonlinear reaction-diffusion equation.     

An alternating direction generalized trapezoidal formula scheme for parabolic differential equations in two space dimensions

A well-known ADI scheme for parabolic differential equations in two space dimensions is the Peaceman-Rachford scheme; this scheme employs the backward Euler formula for integration in time and is unconditionally stable. An ADI Crank -Nicolson scheme, which employs the classical trapezoidal formula for integration in time, is unconditionally unstable. We investigaan ADI implementation of the generalized trapezoidal formula GTF(α) for integration in time. The obtained ADI-GTF(α) scheme is unconditionally stable for all α ≥ 1; interestingly, ADIGTF(α) scheme includes the Peaceman-Rachford scheme for α→∞. Numerical experiments demonstrate that while the Peaceman-Rachford scheme can give quite pronounced unwanted oscillations in the computed solution, an ADI-GTF(α) scheme can provide a more stable and accurate approximation for the true solution.     

Delay‐dependent stability of nonlinear hybrid neutral stochastic differential equations with multiple delays

In this article, we study the delay‐dependent stability of a class of hybrid neutral stochastic differential equations (NSDEs) with multiple delays which is highly nonlinear. Some novel stability criteria are investigated according to Lyapunov functions and generalized Itô's formula. In particular, we generalize and improve the results in Shen et al, Systems & Control Letters (2018) from a single delay to multiple delays.     

Compact alternating direction implicit method to solve two-dimensional nonlinear delay hyperbolic differential equations

In this article, a high-order compact alternating direction implicit method combined with a Richardson extrapolation technique is developed to solve a class of two-dimensional nonlinear delay hyperbolic differential equations. The solvability, stability and convergence of the method are analysed simultaneously in L²- and H¹-norms by the discrete energy method. Numerical experiments are provided to demonstrate the accuracy and efficiency of the schemes.     

Numerical simulation of two-dimensional sine-Gordon solitons via a split cosine scheme

This paper proposes a split cosine scheme for simulating solitary solutions of the sine-Gordon equation in two dimensions, as it arises, for instance, in rectangular large-area Josephson junctions. The dispersive nonlinear partial differential equation allows for soliton-type solutions, a ubiquitous phenomenon in a large variety of physical problems. The semidiscretization approach first leads to a system of second-order nonlinear ordinary differential equations. The system is then approximated by a nonlinear recurrence relation which involves a cosine function. The numerical solution of the system is obtained via a further application of a sequential splitting in a linearly implicit manner that avoids solving the nonlinear scheme at each time step and allows an efficient implementation of the simulation in a locally one-dimensional fashion. The new method has potential applications in further multi-dimensional nonlinear wave simulations. Rigorous analysis is given for the numerical stability. Numerical demonstrations for colliding circular solitons are given.     

Global stability for separable nonlinear delay differential equations

Consider the following separable nonlinear delay differential equation

, where we assume that, there is a strictly monotone increasing function f(x) on (−∞, +∞) such that

In this paper, to the above separable nonlinear delay differential equation, we establish conditions of global asymptotic stability for the zero solution. In particular, for a special wide class of f(x) which contains a case of f(x) = e^x−1, we give more explicit conditions. Applying these, we offer conditions of global asymptotic stability for solutions of nonautonomous logistic equations with delays.     

A numerical algorithm for computation modelling of 3D nonlinear wave equations based on exponential modified cubic B-spline differential quadrature method

In this paper, the authors proposed a method based on exponential modified cubic B-spline differential quadrature method (Expo-MCB-DQM) for the numerical simulation of three dimensional (3D) nonlinear wave equations subject to appropriate initial and boundary conditions. This work extends the idea of Tamsir et al. [An algorithm based on exponential modified cubic B-spline differential quadrature method for nonlinear Burgers’ equation, Appl. Math. Comput. 290 (2016), pp. 111–124] for 3D nonlinear wave type problems. Expo-MCB-DQM transforms the 3D nonlinear wave equation into a system of ordinary differential equations (ODEs). To solve the resulting system of ODEs, an optimal five stage and fourth-order strong stability preserving Runge–Kutta (SSP-RK54) scheme is used. Stability analysis of the proposed method is also discussed and found that the method is conditionally stable. Four test problems are considered in order to demonstrate the accuracy and efficiency of the algorithm.     

Generalized Trapezoidal Formulas for the Symmetric Heat Equation in Polar Coordinates

This paper has two objectives. We first describe one-step time integration schemes for the symmetric heat equation in polar coordinates: u t = v ( u rr +( a / r ) u r ) based on the generalized trapezoidal formulas (GTF( f ) of Chawla et al. [2]. This includes the case of cylindrical symmetry for a =1 and of spherical symmetry for a =2. The obtained GTF( f ) time integration schemes are second order in time and unconditionally stable. We then introduce generalized finite Hankel transforms to obtain an analytical solution of the heat equation for all a S 1, with Dirichlet and Neumann type boundary conditions. Numerical experiments are provided to compare the accuracy and stability of the obtained GTF( f ) time integration schemes with the schemes based on the backward Euler, the classical arithmetic-mean trapezoidal formula and a third order time integration scheme.     

A new operator splitting method for the numerical solution of partial differential equations II

We extend the applications of a new method for splitting operators in partial differential equations introduced by us (A. Rouhi and J. Wright, A new operator splitting method for the numerical solution of partial differential equations, Comput. Phys. Commun. 85 (1995) 18–28, and Spectral implementation of a new operator splitting method for solving partial differential equations, Comput. Phys. (1995), to be published.) to equations in two spatial dimensions, and show how the method allows the use of explicit time stepping methods in some instances when other methods require implicit time stepping. This odd-even splitting method also enables one to increase the order of accuracy of time stepping in a straightforward manner. Our main examples will be the two-dimensional Navier-Stokes equations and the shallow water equations. In the first example we show how the pressure term can be dealt with in simple geometries. We will then discuss the treatment of the diffusion term. Next we will discuss how fast waves can be treated by explicit methods using the odd-even splitting, while retaining all stability and accuracy advantages of usual implicit methods. Our example here will be the shallow water equations in two dimensions.     

Klassische Runge-Kutta-Nyström-Formeln mit Schrittweiten-Kontrolle für Differentialgleichungen\ddot x = f(t,x)

New explicit Runge-Kutta-Nyström formulas for differential equations of the type \(\ddot x = f(t, x)\) are presented. These formulas include a stepsize control procedure based on a complete coverage of the leading term of the local truncation error inx. The formulas require fewer evaluations per step than our Runge-Kutta formulas for first-order differential equations. A numerical example is presented. For results of the same accuracy, the computer time for the new formulas is only about 25% to 50% of the time for the corresponding Runge-Kutta formulas.     

New L-Stable Modified Trapezoidal Methods For The Initial Value Problems

New L-stable trapezoidal formulas obtained by modifying the nonlinear trapezoidal formulas are presented. Numerical results are presented to confirm the theoretical stability analysis.     

L1 Fourier spectral methods for a class of generalized two-dimensional time fractional nonlinear anomalous diffusion equations

In this paper, L1 Fourier spectral methods are derived to obtain the numerical solutions for a class of generalized two-dimensional time-fractional nonlinear anomalous diffusion equations involving Caputo fractional derivative. Firstly, we establish the L1 Fourier Galerkin full discrete and L1 Fourier collocation schemes with Fourier spectral discretization in spatial direction and L1 difference method in temporal direction. Secondly, stability and convergence for both Galerkin and collocation approximations are proved. It is shown that the proposed methods are convergent with spectral accuracy in space and $(2?\alpha )$ order accuracy in time. For implementation, the equivalence between pseudospectral method and collocation method is discussed. Furthermore, a numerical algorithm of Fourier pseudospectral method is developed based on two-dimensional fast Fourier transform (FFT2) technique. Finally, numerical examples are provided to test the theoretical claims. As is shown in the numerical experiments, Fourier spectral methods are powerful enough with excellent efficiency and accuracy.     

Time Stepping Via One-Dimensional Padé Approximation

The numerical solution of time-dependent ordinary and partial differential equations presents a number of well known difficulties—including, possibly, severe restrictions on time-step sizes for stability in explicit procedures, as well as need for solution of challenging, generally nonlinear systems of equations in implicit schemes. In this note we introduce a novel class of explicit methods based on use of one-dimensional Padé approximation. These schemes, which are as simple and inexpensive per time-step as other explicit algorithms, possess, in many cases, properties of stability similar to those offered by implicit approaches. We demonstrate the character of our schemes through application to notoriously stiff systems of ODEs and PDEs. In a number of important cases, use of these algorithms has resulted in orders-of-magnitude reductions in computing times over those required by leading approaches.     

Third-order temporal discrete scheme for the non-stationary Navier–Stokes equations

In this article, we suggest a new third-order time discrete scheme for the two-dimensional non-stationary Navier–Stokes equations. After presenting the Galerkin finite element approximation for the spatial discretization, we consider an implicit/explicit time discrete scheme for the problem, which is based on the two-step Adams–Moulton scheme (implicit scheme) for the linear term and the three-step Adams–Bashforth scheme (explicit scheme) for the nonlinear term. In this method, we only need to solve a linearized discrete system at each time step, so the scheme can converge fast and the computational cost can be reduced. Moreover, under some assumptions, we deduce the stability and optimal error estimate for the velocity in L ²-norm.     

Adaptive Nyström-Runge-Kutta-Methoden für gewöhnliche Differentialgleichungssysteme zweiter Ordnung

A class of generalized Nyström-methods is derived for second order differential equations without first derivatives. These methods are based on local linearization of the system of differential equations. An infinite interval of stability for linear implicit methods is achieved by appropriate choice of the stability matrix. The linear implicit methods are suitable for the integration of large systems of ordinary differential equations resulting from the semi-discretization of hyperbolic differential equations of second order.     

Building Blocks for Arbitrary High Order Discontinuous Galerkin Schemes

In this article we propose the use of the ADER methodology of solving generalized Riemann problems to obtain a numerical flux, which is high order accurate in time, for being used in the Discontinuous Galerkin framework for hyperbolic conservation laws. This allows direct integration of the semi-discrete scheme in time and can be done for arbitrary order of accuracy in space and time. The resulting fully discrete scheme in time does not need more memory than an explicit first order Euler time-stepping scheme. This becomes possible because of an extensive use of the governing equations inside the numerical scheme itself via the so-called Cauchy–Kovalewski procedure. We give an efficient algorithm for this procedure for the special case of the nonlinear two-dimensional Euler equations. Numerical convergence results for the nonlinear Euler equations results up to 8th order of accuracy in space and time are shown