Two-derivative Runge-Kutta Methods Research Articles

Construction of Two-Derivative Runge–Kutta Methods of Order Six

Two-Derivative Runge–Kutta methods have been proposed by Chan and Tsai in 2010 and order conditions up to the fifth order are given. In this work, for the first time, we derive order conditions for order six. Simplifying assumptions that reduce the number of order conditions are also given. The procedure for constructing sixth-order methods is presented. A specific method is derived in order to illustrate the procedure; this method is of the sixth algebraic order with a reduced phase-lag and amplification error. For numerical comparison, five well-known test problems have been solved using a seventh-order Two-Derivative Runge–Kutta method developed by Chan and Tsai and several Runge–Kutta methods of orders 6 and 8. Diagrams of the maximum absolute error vs. computation time show the efficiency of the new method.

Explicit strong stability preserving second derivative multistep methods for the Euler and Navier–Stokes equations

In this paper, we develop the explicit second derivative multistep methods (SDMMs) for the Euler and Navier–Stokes equations. The SDMMs use the calculated results from the previous steps, which can improve computational efficiency. The third-order and fourth-order SDMMs temporal discretization methods are constructed based on the order conditions. Additionally, the strong stability preserving (SSP) condition for the SDMMs is proposed, which provides the corresponding parameters for optimal SSP SDMMs. For numerical experiments, the SDMMs can achieve the order of accuracy as designed on the smooth regions, and the optimal SSP coefficient for the SDMMs is shown to be roughly the same in theory as in the numerical results. Moreover, the SDMMs have approximately twice the computational efficiency compared with the third-order and fourth-order two-derivative Runge–Kutta methods, and the SDMMs are also more efficient than the classical three-stage third-order SSP Runge–Kutta method.

Exponentially fitted two-derivative DIRK methods for oscillatory differential equations

In this work, we construct and derive a new class of exponentially fitted two-derivative diagonally implicit Runge–Kutta (EFTDDIRK) methods for the numerical solution of differential equations with oscillatory solutions. First, a general format of so-called modified two-derivative diagonally implicit Runge–Kutta methods (TDDIRK) is proposed. Their order conditions up to order six are derived by introducing a set of bi-colored rooted trees and deriving new elementary weights. Next, we build exponential fitting conditions in order for these modified TDDIRK methods to treat oscillatory solutions, leading to EFTDDIRK methods. In particular, a family of 2-stage fourth-order, a fifth-order, and a 3-stage sixth-order EFTDDIRK schemes are derived. These can be considered as superconvergent methods. The stability and phase-lag analysis of the new methods are also investigated, leading to optimized fourth-order schemes, which turn out to be much more accurate and efficient than their non-optimized versions. Finally, we carry out numerical experiments on some oscillatory test problems. Our numerical results clearly demonstrate the accuracy and efficiency of the newly derived methods when compared with existing trigonometically/exponentially fitted implicit Runge–Kutta methods and two-derivative Runge–Kutta methods of the same order in the literature.

High Order Two-Derivative Runge-Kutta Methods with Optimized Dispersion and Dissipation Error

In this work we consider explicit Two-derivative Runge-Kutta methods of a specific type where the function f is evaluated only once at each step. New 7th order methods are presented with minimized dispersion and dissipation error. These are two methods with constant coefficients with 5 and 6 stages. Also, a modified phase-fitted, amplification-fitted method with frequency dependent coefficients and 5 stages is constructed based on the 7th order method of Chan and Tsai. The new methods are applied to 4 well known oscillatory problems and their performance is compared with the methods in that of Chan and Tsai.The numerical experiments show the efficiency of the derived methods.

Optimized low-dispersion and low-dissipation two-derivative Runge–Kutta method for wave equations

The paper is devoted to the optimization of the explicit two-derivative sixth-order Runge–Kutta method in order to obtain low dissipation and dispersion errors. The method is dependent on two free parameters, used for the optimization. The optimized method demonstrates the lowest dispersion error in comparison with other widely-used high-order Runge–Kutta methods for hyperbolic problems. The efficiency of the method is demonstrated on the solutions of problems for five linear and nonlinear partial differential equations by the method-of-lines. Spatial derivatives are discretized by finite differences and Petrov–Galerkin approximations. The work-precision and error—CPU time plots, as the dependence of CPU time on space grid resolution, are considered. Also, the optimized method compared with other two-derivative methods. In most examples, the optimized method has better properties, especially for the cases of computations with adaptive stepsize control at high accuracy. The lowest CPU time takes place for the optimized method, especially in the cases of fine space grids.

Trigonometrically fitted fifth-order explicit two-derivative Runge-Kutta method with FSAL property

A new trigonometrically fitted two-derivative explicit Runge-Kutta (TFTDRK) method of order five with FSAL property for solving system of first-order ordinary differential equations (ODEs) with oscillatory solutions are derived. The new method is derived using the property of First Same As Last (FSAL). This method has the advantageous to merge totally first-order ordinary differential systems which their solutions are linear composition of the set of functions {e (u );e (−u)}, or equivalently {s (u );c (u)} when u > 0 is the dominant frequency of the problem. We analyzed the stabilityof our method. The numerical results are presented to illustrate the competence of TFTDRK method compared with some well-known TFRK methods.

A new embedded 4(3) pair of modified two-derivative Runge–Kutta methods with FSAL property for the numerical solution of the Schrödinger equation

A new embedded 4(3) pair of modified two-derivative Runge–Kutta (TDRK) methods with First Same As Last (FSAL) property for the numerical solution of the Schrodinger equation is constructed in this paper. Both the error analysis and phase properties indicate good accuracy of the new pair especially for large eigenvalues. An application to the well-known Lennard-Jones potential confirms the theory and shows that the new pair is more efficient than some high-quality Runge–Kutta(–Nystrom) pairs in the literature.

Modified two-derivative Runge–Kutta methods for the Schrödinger equation

A family of modified two-derivative Runge–Kutta (MTDRK) methods for the integration of the Schrodinger equation are obtained. Two new three-stage and fifth order TDRK methods are derived. The numerical results in the integration of the radial Schrodinger equation with the Woods–Saxon potential are reported to show the high efficiency of our new methods. The results of the error analysis are illustrated by the resonance problem.

Two-derivative Runge–Kutta methods with increased phase-lag and dissipation order for the Schrödinger equation

Optimized explicit two-derivative Runge–Kutta methods with increased phase-lag and dissipation order for the numerical integration of the Schrodinger equation are constructed. The methods are of order five with increased order of phase and dissipation. Numerical results are reported to show the efficiency of the new methods.

Phase-Fitted and Amplification-Fitted Higher Order Two-Derivative Runge-Kutta Method for the Numerical Solution of Orbital and Related Periodical IVPs

A phase-fitted and amplification-fitted two-derivative Runge-Kutta (PFAFTDRK) method of high algebraic order for the numerical solution of first-order Initial Value Problems (IVPs) which possesses oscillatory solutions is derived. We present a sixth-order four-stage two-derivative Runge-Kutta (TDRK) method designed using the phase-fitted and amplification-fitted property. The stability of the new method is analyzed. The numerical experiments are carried out to show the efficiency of the derived methods in comparison with other existing Runge-Kutta (RK) methods.

Derivation of three-derivative Runge-Kutta methods

We introduce an algorithm for a numerical integration of ordinary differential equations in the form of yź = f(y). We extend the two-derivative Runge-Kutta methods (Chan and Tsai, Numer. Algor. 53, 171---194, 2010) to three-derivative Runge-Kutta methods by including the third derivative yźźź=ăź(y)=fźź(y)(f(y),f(y))+fź(y)fź(y)f(y)$y^{\prime \prime \prime }=\hat {g}(y)=f^{\prime \prime }(y)(f(y), f(y))+f^{\prime }(y)f^{\prime }(y)f(y)$. We present an approach based on the algebraic theory of Butcher (Math. Comp. 26, 79---106, 1972) and the źź$\mathcal {B}-$ series theory of Hairer and Wanner (Computing 13, 1---15 (1974)) combined with the methodology of Chan and Chan (Computing 77(3), 237---252, 2006). In this study, special explicit three-derivative Runge-Kutta methods that possess one evaluation of first derivative, one evaluation of second derivative, and many evaluations of third derivative per step are introduced. Methods with stages up to six and of order up to ten are presented. The numerical calculations have been performed on some standard problems and comparisons made with the accessible methods in the literature.

Novel Exponentially Fitted Two-Derivative Runge-Kutta Methods with Equation-Dependent Coefficients for First-Order Differential Equations

The construction of exponentially fitted two-derivative Runge-Kutta (EFTDRK) methods for the numerical solution of first-order differential equations is investigated. The revised EFTDRK methods proposed, with equation-dependent coefficients, take into consideration the errors produced in the internal stages to the update. The local truncation errors and stability of the new methods are analyzed. The numerical results are reported to show the accuracy of the new methods.

Exponentially Fitted Two-Derivative Runge-Kutta Methods for Simulation of Oscillatory Genetic Regulatory Systems.

Oscillation is one of the most important phenomena in the chemical reaction systems in living cells. The general purpose simulation algorithms fail to take into account this special character and produce unsatisfying results. In order to enhance the accuracy of the integrator, the second-order derivative is incorporated in the scheme. The oscillatory feature of the solution is captured by the integrators with an exponential fitting property. Three practical exponentially fitted TDRK (EFTDRK) methods are derived. To test the effectiveness of the new EFTDRK methods, the two-gene system with cross-regulation and the circadian oscillation of the period protein in Drosophila are simulated. Each EFTDRK method has the best fitting frequency which minimizes the global error. The numerical results show that the new EFTDRK methods are more accurate and more efficient than their prototype TDRK methods or RK methods of the same order and the traditional exponentially fitted RK method in the literature.

L(α)-stable variable-order implicit second derivative Runge-Kutta methods

This paper considers the extension of the popular Runge-Kutta methods (RKM) to second derivative Runge-Kutta methods (SDRKMs) for the direct solution of stiff initial value problems (IVPs) of ordinary differential equations (ODEs). The methods are based on using collocation and interpolation technique. The last stage of the input approximation is identical to the output method. The SDRKMs are L(α)-stable for the methods examined. Numerical experiments are given comparing one of these methods with a two derivative Runge Kutta method (TDRKM) and a second derivative linear multistep method (SDLMM).

Two-derivative Runge–Kutta methods for PDEs using a novel discretization approach

We develop a novel and general approach to the discretization of partial differential equations. This approach overcomes the rigid restriction of the traditional method of lines (MOL) and provides flexibility in the treatment of spatial discretization. This method is essential for developing efficient numerical schemes for PDEs based on two-derivative Runge–Kutta (TDRK) methods, where the first and second derivatives must be discretized in an efficient way. This is unlikely to be achieved by using MOL. We then apply the explicit TDRK methods to the advection equations and analyze the numerical stability in the linear advection equation case. We conduct numerical experiments on the Burgers' equation using the TDRK methods developed. We also apply a two-stage semi-implicit TDRK method of order-4 and stage-order-4 to the heat equation. A very significant improvement in the efficiency of this TDRK method is observed when compared to the popular Crank-Nicolson method. This paper is partially based on the work in Tsai's PhD thesis (2011) [10].

Trigonometrically fitted two-derivative Runge-Kutta methods for solving oscillatory differential equations

Explicit trigonometrically fitted two-derivative Runge-Kutta (TFTDRK) methods solving second-order differential equations with oscillatory solutions are constructed. When the second derivative is available, TDRK methods can attain one algebraic order higher than Runge-Kutta methods of the same number of stages. TFTDRK methods have the favorable feature that they integrate exactly first-order systems whose solutions are linear combinations of functions from the set { exp ( i ? x ) , exp ( ? i ? x ) } $\{\exp ({\rm i}\omega x),\exp (-{\rm i}\omega x)\}$ or equivalently the set { cos ( ? x ) , sin ( ? x ) } $\{\cos (\omega x),\sin (\omega x)\}$ with ? > 0 $\omega >0$ the principal frequency of the problem. Four practical TFTDRK methods are constructed. Numerical stability and phase properties of the new methods are examined. Numerical results are reported to show the robustness and competence of the new methods compared with some highly efficient methods in the recent literature.

New optimized two-derivative Runge-Kutta type methods for solving the radial Schrödinger equation

This paper focuses on adapted two-derivative Runge-Kutta (TDRK) type methods for solving the Schrodinger equation. Two new TDRK methods are derived by nullifying their phase-lags and the first derivatives of the phase-lags. Error analysis is carried out by means of asymptotic expressions of the local errors. Numerical results are reported to show the efficiency and robustness of the new methods in comparison with some RK type methods specially tuned to the integration of the radial time-independent Schrodinger equation with the Woods–Saxon potential.

A New Trigonometrically Fitted Two-Derivative Runge-Kutta Method for the Numerical Solution of the Schrödinger Equation and Related Problems

A new trigonometrically fitted fifth-order two-derivative Runge-Kutta method with variable nodes is developed for the numerical solution of the radial Schrödinger equation and related oscillatory problems. Linear stability and phase properties of the new method are examined. Numerical results are reported to show the robustness and competence of the new method compared with some highly efficient methods in the recent literature.

On explicit two-derivative Runge-Kutta methods

The theory of Runge-Kutta methods for problems of the form y′ = f(y) is extended to include the second derivative y′′ = g(y): = f′(y)f(y). We present an approach to the order conditions based on Butcher’s algebraic theory of trees (Butcher, Math Comp 26:79–106, 1972), and derive methods that take advantage of cheap computations of the second derivatives. Only explicit methods are considered here where attention is given to the construction of methods that involve one evaluation of f and many evaluations of g per step. Methods with stages up to five and of order up to seven including some embedded pairs are presented. The first part of the paper discusses a theoretical formulation used for the derivation of these methods which are also of wider applicability. The second part presents experimental results for non-stiff and mildly stiff problems. The methods include those with the computation of one second derivative (plus many first derivatives) per step, and embedded methods for changing stepsize as well as those involving one first derivative (plus many second derivatives) per step. The experiments have been performed on standard problems and comparisons made with some standard explicit Runge-Kutta methods.