Showing papers on "L-stability published in 2011"

ADER Schemes for Nonlinear Systems of Stiff Advection---Diffusion---Reaction Equations

Abstract: In this article we extend the high order ADER finite volume schemes introduced for stiff hyperbolic balance laws by Dumbser, Enaux and Toro (J. Comput. Phys. 227:3971---4001, 2008) to nonlinear systems of advection---diffusion---reaction equations with stiff algebraic source terms. We derive a new efficient formulation of the local space-time discontinuous Galerkin predictor using a nodal approach whose interpolation points are tensor-products of Gauss---Legendre quadrature points. Furthermore, we propose a new simple and efficient strategy to compute the initial guess of the locally implicit space-time DG scheme: the Gauss---Legendre points are initialized sequentially in time by a second order accurate MUSCL-type approach for the flux term combined with a Crank---Nicholson method for the stiff source terms. We provide numerical evidence that when starting with this initial guess, the final iterative scheme for the solution of the nonlinear algebraic equations of the local space-time DG predictor method becomes more efficient. We apply our new numerical method to some systems of advection---diffusion---reaction equations with particular emphasis on the asymptotic preserving property for linear model systems and the compressible Navier---Stokes equations with chemical reactions.

Exponential Runge-Kutta Methods for Stiff Kinetic Equations

Abstract: We introduce a class of exponential Runge-Kutta integration methods for kinetic equations. The methods are based on a decomposition of the collision operator into an equilibrium and a nonequilibrium part and are exact for relaxation operators of BGK type. For Boltzmann-type kinetic equations they work uniformly for a wide range of relaxation times and avoid the solution of nonlinear systems of equations even in stiff regimes. We give sufficient conditions in order that such methods are unconditionally asymptotically stable and asymptotic preserving. Such stability properties are essential to guarantee the correct asymptotic behavior for small relaxation times. The methods also offer favorable properties such as nonnegativity of the solution and entropy inequality. For this reason, as we will show, the methods are suitable both for deterministic as well as probabilistic numerical techniques.

Galerkin and Runge–Kutta methods: unified formulation, a posteriori error estimates and nodal superconvergence

Abstract: We unify the formulation and analysis of Galerkin and Runge–Kutta methods for the time discretization of parabolic equations. This, together with the concept of reconstruction of the approximate solutions, allows us to establish a posteriori superconvergence estimates for the error at the nodes for all methods.

Explicit stabilized Runge-Kutta methods

Abstract: Explicit Runge-Kutta methods with the stability domains extended along the real axis are examined. For these methods, a simple and efficient procedure for calculating the stability polynomials is proposed. Three techniques for constructing methods with given stability polynomials are considered. Methods of the second and third orders are constructed, and their accuracy as applied to solving the Prothero-Robinson equation is examined. A comparison of the above methods on some test problems is performed.

Strang‐type preconditioners applied to ordinary and neutral differential‐algebraic equations

Abstract: This paper deals with boundary-value methods (BVMs) for ordinary and neutral differential-algebraic equations. Different from what has been done in Lei and Jin (Lecture Notes in Computer Science, vol. 1988. Springer: Berlin, 2001; 505–512), here, we directly use BVMs to discretize the equations. The discretization will lead to a nonsymmetric large-sparse linear system, which can be solved by the GMRES method. In order to accelerate the convergence rate of GMRES method, two Strang-type block-circulant preconditioners are suggested: one is for ordinary differential-algebraic equations (ODAEs), and the other is for neutral differential-algebraic equations (NDAEs). Under some suitable conditions, it is shown that the preconditioners are invertible, the spectra of the preconditioned systems are clustered, and the solution of iteration converges very rapidly. The numerical experiments further illustrate the effectiveness of the methods. Copyright © 2011 John Wiley & Sons, Ltd.

Implicit Two Step Continuous Hybrid Block Methods with Four Off-Steps Points for Solving Stiff Ordinary Differential Equation

Abstract: In this paper, a self starting two step continuous block hybrid formulae (CBHF) with four Off-step points is developed using collocation and interpolation procedures. The CBHF is then used to produce multiple numerical integrators which are of uniform order and are assembled into a single block matrix equation. These equations are simultaneously applied to provide the approximate solution for the stiff ordinary differential equations. The order of accuracy and stability of the block method is discussed and its accuracy is established numerically. Keywords— Collocation and Interpolation, Continuous Hybrid Block Formulae, Off-Step Points, Stability, Stiff ODEs.

Fifth order two-point block backward differentiation formulas for solving ordinary differential equations.

Abstract: The new method derived is called the fifth order two-point Block Backward Differentiation Formulas (BBDF(5)) method for solving first order stiff Ordinary Differential Equations (ODEs). This method possesses the requirement for stiffly stable and suitable to solve stiff problems. We also discussed further the implementation of the method using Newton Iteration. The numerical results are presented to verify the efficiency of this method as compared to the Classical Backward Differentiation Formula (BDF) method and ode15s in Matlab. The BBDF(5) method outperformed the BDF method and ode15s in terms of maximum error and execution time.

Stability analysis of Runge–Kutta methods for differential equations with piecewise continuous arguments of mixed type

Abstract: This paper deals with the stability analysis of the Runge–Kutta methods for a differential equation with piecewise continuous arguments of mixed type The stability regions of the analytical solution are given The necessary and sufficient conditions under which the numerical solution is asymptotically stable are discussed The conditions under which the analytical stability region is contained in the numerical stability region are obtained and some numerical experiments are given

A ($\alpha $)-Stable Linear Multistep Methods for Stiff IVPs in ODEs

Abstract: In this paper, a class of A(α)-stable linear multistep formulas for stiff initial value problems (IVPs) in ordinary differential equations (ODEs) is developed. The boundary locus of the methods shows that the schemes are A-stable for step number k ≤ 3 and stiffly stable for k =4 ,5 and 6. Some numerical results are reported to illustrate the method.

Integration of chemical stiff ODEs using exponential propagation method

Abstract: In this paper, we study the numerical long time integration of large stiff systems of differential equations arising from chemical reactions by exponential propagation methods. These methods, which typically converge faster, use matrix-vector products with the exponential or other related function of the Jacobian that can be effectively approximated by Krylov sabspace methods. We equip these methods to an automatic stepsize control technique and apply the method of order 4 for numerical integration of some famous stiff chemical problems such as Belousov-Zhabotinskii reaction, the Chapman atmosphere, Hydrogen chemistry, chemical Akzo-Nobel problem and air pollution problem.

An eighth order Backward Differentiation Formula with Continuous Coefficients for Stiff Ordinary Differential Equations

Abstract: A block backward differentiation formula of uniform order eight is proposed for solving first order stiff initial value problems (IVPs). The conventional 8-step Backward Differentiation Formula (BDF) and additional methods are obtained from the same continuous scheme and assembled into a block matrix equation which is applied to provide the solutions of IVPs on non-overlapping intervals. The stability analysis of the method indicates that the method is L0-stable. Numerical results obtained using the proposed new block form show that it is attractive for solutions of stiff problems and compares favourably with existing ones. Keywords—Stiff IVPs, System of ODEs,Backward differentiation formulas, Block methods, Stability.

Hybrid Extended Backward Differentiation Formulas for Stiff Systems

Abstract: In this paper we present details of a new class of hybrid methods which are based on backward differentiation formula (BDF) for the numerical solution of ordinary differential equations. In these methods, the first derivative of the solution in one super future point as well as in one off-step point is used to improve the absolute stability regions. The constructed methods are A( )-stable up to order 9 so that, as it is shown in the numerical experiments, they are superior for stiff systems.

Relative Global Error Control in the RKQ Algorithm for Systems of Ordinary Differential Equations

Abstract: We generalize the RKrvQz algorithm to solve nonstiff initial-value problems in ordinary differential equations. The algorithm can now be applied to systems of nonstiff initial-value problems (IVPs) in ordinary differential equations, and both relative error and absolute error can be controlled, locally and globally. We demonstrate the algorithm by solving the simple harmonic oscillator for moderate and strict tolerances.

Class 2+1 Hybrid BDF-Like methods for the numerical solutions of ordinary differential equations

Abstract: In this article, the details of new hybrid methods have been presented to solve systems of ordinary differential equations (ODEs). These methods are based on backward differentiation formulae (BDF) where one additional stage point (or off-step point) and two step points have been used in the first derivative of the solution to improve the absolute stability regions compared with some existing methods such as BDF, extended BDF (EBDF) and modified EBDF (MEBDF). Stability domains of our new methods have been obtained showing that these methods, we say Class 2+1 Hybrid BDF-Like methods, are A-stable for order p, p=3,4, and A(?)-stable for order p, p=5, 6, 7, 8. Numerical results are also given for five test problems.

Some Third Order Methods for Solving Systems of Nonlinear Equations

Abstract: Based on Traub's methods for solving nonlinear equation f(x) = 0, we develop two families of third-order methods for solving system of nonlinear equations F(x) = 0. The families include well-known existing methods as special cases. The stability is corroborated by numerical results. Comparison with well-known methods shows that the present methods are robust. These higher order methods may be very useful in the numerical applications requiring high precision in their computations because these methods yield a clear reduction in number of iterations. Keywords—Nonlinear equations and systems, Newton's method, fixed point iteration, order of convergence.

Explicit Numerical Methods for Solving Stiff Dynamical Systems

Abstract: The precise integration method can give precise numerical result for linear invariant dynamical system, and can be used to solve stiff linear invariant dynamical system. In this paper, precise integration is compounded with Runge–Kutta method and a new effective integration method is presented for solving nonlinear stiff problems. The numerical examples are given to demonstrate the validity and effectiveness of the proposed method.

Explicit adaptive Runge-Kutta methods for stiff and oscillation problems

Abstract: Explicit Runge-Kutta methods with the coefficients tuned to the problem of interest are examined. The tuning is based on estimates for the dominant eigenvalues of the Jacobian matrix obtained from the results of the preliminary stages. Test examples demonstrate that methods of this type can be efficient in solving stiff and oscillation problems.

A New Numerical Method for Solving Convection-Diffusion Equations

Abstract: In this paper, we using semi-discrete method, transformed convection-diffusion equation into a ODEs: \(\frac{dU(t)}{dt}\) = AU(t), then we get the solution of the ODEs: U(t) = e tA U0. Furthermore, we give a numerical approximation for e tA and get a special difference scheme for solving the convection-diffusion equation which improve the accuracy order and stability condition greatly. The accuracy order is fourth order and second order in space and time direction respectively. Finally, numerical result shows that this method is effective.

The Modified Variational Iteration Method for Solving Linear and Nonlinear Ordinary Differential Equations

Abstract: In this work, the modified variational iteration method (MVIM) is applied to solve linear and nonlinear ordinary differential equations such as Lane-Emden, Emden-Fowler and Riccati equations. The MVIM provides a sequence of functions which is convergent to the exact solution and is capable to cancel some of the repeated calculations and reduce the cost of operation in comparison with VIM. The method is very simple and easy.

Three-stage Stiffly Accurate Runge-Kutta Methods for Stiff Stochastic Differential Equations

Abstract: In this paper we discuss diagonally implicit and semi-implicit methods based on the three-stage stiffly accurate Runge-Kutta methods for solving Stratonovich stochastic differential equations(SDEs).Two methods,a three-stage stiffly accurate semi-implicit(SASI3) method and a three-stage stiffly accurate diagonally implicit (SADI3) method,are constructed in this paper.In particular,the truncated random variable is used in the implicit method.The stability properties and numerical results show the effectiveness of these methods in the pathwise approximation of stiff SDEs.

Special functions in academician A.A. Dorodnicyn’s scientific legacy: Special functions arising in the solution of boundary value problems for second-order singular ordinary differential equations by asymptotic methods

Abstract: On the occasion of the centenary of Academician A.A. Dorodnicyn’s birth, his main results concerning asymptotic methods for solving second-order singular ordinary differential equations and special functions related to these equations are presented (with certain corrections and additions).

L-устойчивый (4,2)-метод четвертого порядка для решения жестких задач

Abstract: (M,k)-methods for solving stiff problems, in which on each step two times the right-hand side of the system of differential equations is calculated are investigated. It is shown that the maximum order of accuracy of the L-stable (m,2)-method is equal to four. (4,2)-method of maximal order is built.

Perturbations of Multipliers of Systems of Periodic Ordinary Differential Equations

Abstract: The paper deals with periodic systems of ordinary differential equations (ODEs). A new approach to the investigation of variations of multipliers under perturbations is suggested. It enables us to establish explicit conditions for the stability and instability of perturbed systems.