Runge-Kutta Type Research Articles

The Algebraic Structure of an Implicit Runge-Kutta Type Method

Abstract: In this paper, the theory of linear transformation (Homomorphism) and monomorphism is applied to a first-order Runge-Kutta Type Method illustrated in a Butcher Table and the extended second order Runge- Runge-Kutta type Method to substantiate their uniform order and error constants obtained. A homomorphism is a mapping from one group to another group which preserves the group operations. It’s sometimes called the operation preserving function. The methods which initially are Linear Multistep were reformulated into Runge-Kutta (R-K) Type to establish the advantages the R-K has over Linear Multistep. The first-order Linear multistep was reformulated into first-order R-K type which was further extended to second order. This extension can be made to higher order. For this study, the extension was limited to the second order.

Embedded Schemes of the Runge-Kutta Type for the Direct Solution of Fourth-Order Ordinary Differential Equations

Embedded Schemes of the Runge-Kutta Type for the Direct Solution of Fourth-Order Ordinary Differential Equations

An exponential stochastic Runge–Kutta type method of order up to 1.5 for SPDEs of Nemytskii-type

Abstract For the approximation of solutions for stochastic partial differential equations, numerical methods that obtain a high order of convergence and at the same time involve reasonable computational cost are of particular interest. We therefore propose a new numerical method of exponential stochastic Runge–Kutta type that allows for convergence with a temporal order of up to $\frac{3}/{2}$ and that can be combined with several spatial discretizations. The developed family of derivative-free schemes is tailored to stochastic partial differential equations of Nemytskii-type, i.e., with pointwise multiplicative noise operators. We prove the strong convergence of these schemes in the root mean-square sense and present some numerical examples that reveal the theoretical results.

High-order schemes of exponential time differencing for stiff systems with nondiagonal linear part

Exponential time differencing methods are a power tool for high-performance numerical simulation of computationally challenging problems in condensed matter physics, fluid dynamics, chemical and biological physics, where mathematical models often possess fast oscillating or decaying modes—in other words, are stiff systems. Practical implementation of these methods for the systems with nondiagonal linear part of equations is exacerbated by infeasibility of an analytical calculation of the exponential of a nondiagonal linear operator; in this case, the coefficients of the exponential time differencing scheme cannot be calculated analytically. We suggest an approach, where these coefficients are numerically calculated with auxiliary problems. We rewrite the high-order Runge–Kutta type schemes in terms of the solutions to these auxiliary problems and practically examine the accuracy and computational performance of these methods for a heterogeneous Cahn–Hilliard equation, a sixth-order spatial derivative equation governing pattern formation in the presence of an additional conservation law, and a Fokker–Planck equation governing macroscopic dynamics of a network of neurons.

A fourth order Runge-Kutta type of exponential time differencing and triangular spectral element method for two dimensional nonlinear Maxwell's equations

In this paper, we study a numerical scheme to solve the nonlinear Maxwell's equations. The discrete scheme is based on the triangular spectral element method (TSEM) in space and the exponential time differencing fourth-order Runge-Kutta (ETDRK4) method in time. TSEM has the advantages of spectral accuracy and geometric flexibility. The ETD method involves exact integration of the linear part of the governing equation followed by an approximation of an integral involving the nonlinear terms. The RK4 scheme is introduced for the time integration of the nonlinear terms. The stability region of the ETDRK4 method is depicted. Moreover, the contour integral in the complex plan is utilized and improved to compute the matrix function required by the implementation of ETDRK4. The numerical results demonstrate that our proposed method is of exponential convergence with the order of basis function in space and fourth order accuracy in time.

A 4-Stage Runge-Kutta Type Method for Solution of Stiff Ordinary Differential Equations

In this paper, a 2 step implicit block hybrid linear multistep method was reformulated into a 4-stage block hybrid Runge-Kutta Type Method via the butcher analysis. The method can be used to solve first order stiff ordinary differential equations. A numerical example solved with the proposed method showed a better result in comparison with an existing method.

Delay-dependent stability of predictor–corrector methods of Runge–Kutta type for stochastic delay differential equations

Delay-dependent stability of predictor–corrector methods of Runge–Kutta type for stochastic delay differential equations

Explicit integrators for nonlocal equations: The case of the Maxey-Riley-Gatignol equation

The Maxey-Riley-Gatignol (MRG) equation, which describes the dynamics of an inertial particle in nonuniform and unsteady flow, is an integro-differential equation with a memory term and its solution lacks a well-defined Taylor series at t = 0 t=0 . In particulate flows, one often seeks trajectories of millions of particles simultaneously, and the numerical solution to the MRG equation for each particle becomes prohibitively expensive due to its ever-rising memory costs. In this paper, we present an explicit numerical integrator for the MRG equation that inherits the benefits of standard time-integrators, namely a constant memory storage cost, a linear growth of operational effort with simulation time, and the ability to restart a simulation with the final state as the new initial condition. The integrator is based on a Markovian embedding of the MRG equation. The integrator and the embedding are consequences of a spectral representation of the solution to the linear MRG equation. We exploit these to extend the work of Cox and Matthews [J. Comput. Phys. 176 (2002), 430–455] and derive Runge-Kutta type iterative schemes of differing orders for the MRG equation. Our approach may be generalized to a large class of systems with memory effects.

Explicit Runge-Kutta Method for Evaluating Ordinary Differential Equations of type v^vi = f(u, v, v′)

The initiative of this paper is to present the Runge Kutta Type technique for the development of mathematical solutions to the problems concerning to ordinary differential equation of order six of structure v vi = f(u, v, v′ ) denoted as RKSD with initial conditions. The three and four stage Runge-Kutta methods with order conditions up to order seven (RKSD7) have been designed to evaluate global and local truncated errors for the ordinary differential equation of order six. The framework and evaluation of equations with their results are well established to obtain the effectiveness of RK method towards implicit function satisfying the required initial conditions and for obtaining zero-stability of RKSD7 in terms of their accuracy with maximum precision under minimal processing.

Implicit Block and Runge- Kutta type Methods for Solution of Second-Order Ordinary Differential Equations.

In this research, implicit discrete schemes which form our block integrators were developed for solving Initial Valued Problems of Ordinary Differential Equations. The equivalent second order Runge-Kutta type Methods (RKTM) were also constructed for the same purpose. Both methods were demonstrated on linear and nonlinear problems of Ordinary Differential Equations. Numerical results obtained from RKTM show that the method is competitive with the existing one.

A four-stage multiderivative explicit Runge-Kutta method for the solution of first order ordinary differential equations

The development of numerical methods for the solution of initial value problems in ordinary differential equation have turned out to be a very rapid research area in recent decades due to the difficulties encountered in finding solutions to some mathematical models composed into differential equations from real life situations. Researchers have in recent times, used higher derivatives in the derivation of numerical methods to produce totally new ways of solving these equations. In this article, a new Runge-Kutta type methods with reduced number of function evaluations in the increment function is constructed, analyzed and implemented. This proposed method border on the use of higher derivatives up to the second derivative in the ki terms of Runge-Kutta method in order to achieve a higher order of accuracy. The qualitative features: local truncation error, consistency, convergence and stability of the new method were investigated and established. Numerical examples were also performed on some initial value problems to confirm the accuracy of the new method and compared with some existing methods of which the numerical results show that the new method competes favorably.

Runge–Kutta type Time Stepping Methods for Space Fractional Reaction Diffusion Model with Restricted Padé Approximation

A new numerical method that solves a time-space-fractional reaction-diffusion equation effectively is presented. The space-fractional Riesz operator is first discretized using fourth order compact finite differences, resulting in a system with a linear stiff term. The system of differential equations is then solved through time integration using an exponential time difference method, which explicitly handles the nonlinear non-stiff term. Restricted Padé rational approximation of a single real pole is used to approximate the matrix exponential e-A. The problems concerning the stability and computational efficiency of this new approach are tackled by means of a splitting technique. Based on compact finite differences with third-order accuracy in time and restricted Padé approximation, this novel efficient technique reduces computing cost and effectively tackles the problems with non-smooth initial data. The superiority of new method in terms of accuracy, reliability, and computational efficiency is finally shown by numerical examples.

Comparative Analysis of Some New Runge-Kutta Type Techniques on the Solution of First Order Initial Value Problem in Ordinary Differential Equations

Comparative Analysis of Some New Runge-Kutta Type Techniques on the Solution of First Order Initial Value Problem in Ordinary Differential Equations

Efficient pricing of options in jump–diffusion models: Novel implicit–explicit methods for numerical valuation

This paper presents novel implicit–explicit Runge–Kutta type methods for numerically simulating partial integro-differential equations that arise when pricing options under jump–diffusion models. These methods offer an alternative approach that avoids the need for numerical or analytical inversion of the coefficient matrix. The pricing of European options is formulated as a partial integro-differential equation, while the pricing of American options are treated as a linear complementarity problem. The developed implicit–explicit Runge–Kutta type method is combined with an operator splitting technique to efficiently solve the linear complementarity problem. Stability and convergence analysis of the proposed methods are established using discrete ℓ2-norm. To validate their efficiency and accuracy, the methods are applied to pricing European and American options under Merton’s and Kou’s models, and the computed results are compared with those reported in the literature.

A hybrid fourth order time stepping method for space distributed order nonlinear reaction-diffusion equations

A new fourth order highly efficient Exponential Time Differencing Runge-Kutta type hybrid time stepping method is developed by hybriding two fourth order methods. The new hybrid method takes advantages of a positivity preserving L-stable method to damp unnatural oscillations due to low regularity of the initial data and of a computationally efficient A-stable method to achieve optimal order convergence. Computational efficiency and stability of the new method is further enhanced by applying a splitting technique which makes it possible to implement the method on parallel processors. A hybriding criteria is presented and an algorithm based on the hybrid method is developed. The method is implemented to solve two two-dimensional problems having Riesz space distributed order diffusion and nonlinear reaction terms, an Allen-Cahn equation with a cubic nonlinearity and an Enzyme Kinetics equation with a rational nonlinearity. Fourth-order temporal convergence is obtained through numerical experiments. A third test problem with exact solution available is also considered and both spatial as well as temporal orders of convergence are computed. High computational efficiency of the method is shown by recording the CPU time. Numerical solutions using different order strength distribution functions are also presented.

Efficient Frequency-Dependent Coefficients of Explicit Improved Two-Derivative Runge-Kutta Type Methods for Solving Third-Order IVPs

This study aims to propose sixth-order two-derivative improved Runge-Kutta type methods adopted with exponentially-fitting and trigonometrically-fitting techniques for integrating a special type of third-order ordinary differential equation in the form u^''' (t)=f(t,u(t),u^' (t)). The procedure of constructing order conditions comprised of a few previous steps, k-i for third-order two-derivative Runge-Kutta-type methods, has been outlined. These methods are developed through the idea of integrating initial value problems exactly with a numerical solution in the form of linear composition of the set functions e^ѡt and e^(-ѡt)for exponentially fitted and e^iѡt and e^(-iѡt) for trigonometrically-fitted with ѡ ϵ R. Parameters of two-derivative Runge-Kutta type method are adapted into principle frequency of exponential and oscillatory problems to construct the proposed methods. Error analysis of proposed methods is analysed, and the computational efficiency of proposed methods is demonstrated in numerical experiments compared to other existing numerical methods for integrating third-order ordinary differential equations with an exponential and periodic solution.

A nine-parametric family of embedded methods of sixth order

In the paper an effective explicit Runge—Kutta type method of the sixth order with an embedded error estimator of order four is presented. The method is applied to the systems that can be structurally partitioned into three subsystems. Its computational scheme effectively uses the structural properties. However this leads to much larger systems of order conditions. These nonlinear conditions and the algorithm of finding a solution with nine free parameters are presented. A certain computational scheme is written down and a numerical comparison to Dormand—Prince pairs of orders 5 and 6 is performed.

Анализ численных методов интегрирования жестких систем высокой размерности в SimInTech

When modeling complex dynamic processes, it becomes necessary to numerically solve the Cauchy problem for systems of ordinary differential equations (ODEs). The efficiency of the applied numerical methods depends on the degree of stiffness and dimension of the problem [1–3]. Depending on the task class, different methods behave differently.This article provides a comparative analysis of explicit adaptive and diagonal-implicit integration methods implemented in the SimInTech software [4]. The SimInTech software package is designed to simulate complex dynamic processes in systems of various classes. The system supports the ability to develop models in the form of block diagrams, as well as describe systems of differential equations using the built-in programming language and simulate event-driven systems and finite automata.It is shown that the most effective in solving problems of the considered class are the diagonal-implicit Runge-Kutta type integration methods – DIRK2 and DIRK4 from the SimInTech package. The DIRK3 method is inferior due to the large number of calculations of the Jacobian matrix. The preferred method is rather The DIRK2 method is preferable in this case, because it has a greater number of time steps with almost the same performance and does not increase the integration step so much with relatively low accuracy settings of the integration method. Of the explicit methods of the Runge-Kutta type with an adaptive numerical scheme, the “Adaptive-5” method is the most effective for solving problems of this class. For problems of this class, we can recommend the use of explicit integration methods “Adaptive-5”, “Adaptive-1” with a small system dimension.The traditional implicit Gear and Euler integration methods also effectively solve this problem, provided that the algorithm for calculating the Jacobian matrix is effectively implemented.

Convection linearization of energy conservative implicit method for incompressible turbulence simulations

Time accurate implicit methods for unsteady turbulent flows are proposed based on the energy conservative finite difference method of the incompressible Navier-Stokes equations with convection linearization. The standard and Runge-Kutta type high-order temporal linearizations are introduced for the convective term. The time increment independent solution of turbulence is explored for the linearized coupled governing equations. Coupled and uncoupled (splitting) algorithms are then considered to isolate the effect of splitting error. The simulation examples are periodic inviscid, Taylor-Green vortex, and DNS of turbulent plane channel flows. These are test cases for the energy conservation property, the order of accuracy, and the time increment independence of turbulence statistics, respectively. The second order accurate two-stage convection linearized semi-implicit Runge-Kutta (SIRK2L) method works very well and has almost the same conservation property as the fully (spatio-temporally) conservative nonlinear method based on the implicit midpoint (IM) time marching. The second place is the standard third order convection linearization (IM3L) method. The second order convection linearization method (IM2L) is not recommended, although it may be preferred by the point of view of formal accuracy. The effect of splitting error is significant, and the numbers of coupling iteration of 5 and 10 are required for SIRK2L and IM3L methods with the uncoupled (splitting) algorithm, respectively, for the turbulent plane channel flow simulation. Finally, the DNS of forward-facing step flow is carried out as an example of practical problems.

New two-derivative implicit-explicit Runge-Kutta methods for stiff reaction-diffusion systems

Reaction-diffusion systems are extensively used in the mathematical modeling of biological and chemical systems to explain the Turing instability. Generally, reaction-diffusion systems are highly stiff in both reaction and diffusion terms. This paper discusses a new class of two-derivative implicit-explicit (IMEX) Runge-Kutta (RK) type methods for the numerical simulations of stiff reaction-diffusion systems. The present methods do not require numerical inversion of the coefficient matrix - computationally explicit. Stability properties of the developed methods are compared with the similar methods discussed in the literature. Moreover, accuracy and efficiency of the developed methods are validated by numerical simulations of spatiotemporal pattern formations for different reaction-diffusion systems, such as, phase-separation, Schnakenberg model and electrodeposition process.