Kutta Type Research Articles

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.

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 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.

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.

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.

New RK type time-integration methods for stiff convection–diffusion–reaction systems

Convection–diffusion–reaction equations (CDREs) are extensively used to model the various physical processes, which are generally stiff in both reaction and diffusion terms. This paper discusses a new class of implicit–explicit Runge–Kutta (IMEX RK) type methods for numerical simulations of the convection–diffusion–reaction systems. The developed approach does not need to invert the coefficient matrix, despite being implicit in nature. The Fourier stability analysis is performed to gauge the properties of the developed methods using one- and two-dimensional scalar CDRE as model systems. Additionally, numerical simulation of the one-dimensional inhomogeneous linear CDRE, contaminant transport with kinetic Langmuir sorption model, and two-dimensional chemotaxis model that incorporates volume-filling effect and nonlinear diffusion are used to validate the effectiveness and robustness of the developed methods. The numerical solutions match the exact solutions discussed in the literature very well.

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.

Time-efficient reformulation of the Lobatto III family of order eight

Implicit block methods for solving initial value problems in ordinary differential equations are well-known among the contemporary scientific community, since they are cost-effective, self-starting, consistent, stable, and usually converge fast when applied to solve particularly stiff models. These characteristics of block methods are the primary reasons for the one-step optimized block method devised in the present research study with three off-grid points. Theoretical analysis, including the order of convergence, consistency, zero-stability, A-stability, order stars, and the local truncation error, are considered. The obtained method may be categorized as the well-known Lobatto IIIA Runge–Kutta method. The superiority of the devised method over various existing approaches having similar features is proved via numerical simulations of stiff and nonlinear differential systems. Furthermore, a suitable reformulation of the devised method results in considerable savings in computation time, as revealed through the efficiency plots. This turns out in a strategy to reformulate Runge–Kutta type methods in order to get a better performance.

The art of solving a large number of non-stiff, low-dimensional ordinary differential equation systems on GPUs and CPUs

This paper discusses the main performance barriers for solving a large number of independent ordinary differential equation systems on processors (CPU) and graphics cards (GPU). With a naïve approach, for instance, the utilisation of a CPU can be as low as 4% of its theoretical peak processing power. The main barriers identified by the detailed analysing of the hardware architectures and profiling using hardware performance monitoring units are as follows. First, exploitation of the SIMD capabilities of the CPU via vector registers. The solution is to implement/enforce explicit vectorisation. Second, hiding instruction latencies on both CPUs and GPUs that can be achieved with increasing (instruction-level) parallelism. Third, the efficient handling of large timescale differences or event handling using the massively parallel architecture of GPUs. A viable option to overcome this difficulty is asynchronous time stepping. The above optimisation techniques and their implementation possibilities are discussed and tested on three program packages: MPGOS written in C++ and specialised only for GPUs; ODEINT implemented in C++, which supports execution on both CPUs and GPUs; finally, DifferentialEquations.jl written in Julia that also supports execution on both CPUs and GPUs. The tested systems (Lorenz equation, Keller–Miksis equation and a pressure relief valve model) are non-stiff and have low dimension. Thus, the performance of the codes are not limited by memory bandwidth, and Runge–Kutta type solvers are efficient and suitable choices. The employed hardware are an Intel Core i7-4820K CPU with 30.4 GFLOPS peak double-precision performance per cores and an Nvidia GeForce Titan Black GPU that has a total of 1707 GFLOPS peak double-precision performance.

Runge–Kutta Type Discrete Circadian RNN for Resolving Tri-Criteria Optimization Scheme of Noises Perturbed Redundant Robot Manipulators

In order to resist periodic interfere in robot hardware or environment, a Runge–Kutta type discrete-time circadian rhythms neural network (RK-DCRNN) model is proposed, and investigated to plan the motion of redundant robot manipulators. To achieve the optimal control, a quadratic programming-based acceleration-level hybrid tri-criteria (ALHT) scheme is first designed, which simultaneously minimize the acceleration norm, torque norm, and joint-angle shift-free indices. Second, according to the neural dynamic design method, a continuous-time circadian rhythms neural network model is exploited, and then based on the Runge–Kutta numerical differential method, a discrete-time circadian rhythms neural network model is obtained. Third, the convergence of the proposed RK-DCRNN model is proved by detailed mathematical derivation. Fourth, comparative simulations and physical experiments verify that the proposed RK-DCRNN model can suppress the accumulation of position error in the motion planning of manipulators.

On modeling column crystallizers and a Hermite predictor–corrector scheme for a system of integro-differential equations

Abstract Multistaged crystallization systems are used in the production of many chemicals. In this article, employing the population balance framework, we develop a model for a column crystallizer where particle agglomeration is a significant growth mechanism. The main part of the model can be reduced to a system of integrodifferential equations (IDEs) of the Volterra type. To solve this system simultaneously, we examine two numerical schemes that yield a direct method of solution and an implicit Runge–Kutta type method. Our numerical experiments show that the extension of a Hermite predictor–corrector method originally advanced in Khanh (1994) for a single IDE is effective in solving our model. The numerical method is presented for a generalization of the model which can be used to study and simulate a number of possible operating profiles of the column.

Methods for solving the initial value problem with a two-sided estimate of the local error

Numerical methods for solving the initial value problem for ordinary differential equations are proposed. Embedded methods of order of accuracy 2(1), 3(2) and 4(3) are constructed. To estimate the local error, two-sided calculation formulas were used, which give estimates of the main terms of the error without additional calculations of the right-hand side of the differential equation, which favorably distinguishes them from traditional two-sided methods of the Runge- Kutta type.

Exponential Time Differencing for Stiff Systems with Nondiagonal Linear Part

Exponential time differencing methods provide instability-free explicit schemes for systems with fast decaying or oscillating modes (stiff systems), without limitation on the time step size. Moreover, with these methods, one can drastically diminish the error accumulation rate for numerical simulation of conservative systems. The methods yield an especially large performance gain for PDEs with high order of spatial derivatives. Simultaneously, the problem of analytical calculation of coefficients of exponential time differencing schemes becomes laborious or unsolvable in the case of a nondiagonal form of the principal linear part of equations. We introduce an approach, where the scheme coefficients are obtained from the direct numerical integration of certain auxiliary problems over a short time interval—one scheme step size. The approach is universal and its implementation is illustrated with four examples: analytically solvable system of two first-order ODEs, one-dimensional reaction-diffusion system under time-dependent conditions, two-dimensional reaction-diffusion system under time-independent and time-dependent conditions, and one-dimensional Cahn–Hilliard equation with constant coefficients. The employment of an exponential time differencing method of the two-step Runge–Kutta type yields a simulation performance gain for the diffusion-type equation, with program optimization made. Without program optimization, the performance gain increases by one order with respect to the spatial step size for each order of the highest spatial derivative, and appears starting from the third order of the derivative. With the tested method, one can extend the study of an analogue of the Anderson localization to two- and three-dimensional active media and achieve an acceptable performance for the direct numerical simulations of dynamics of the probability density function for active Brownian particles.

THE ORDER AND ERROR CONSTANT OF A RUNGE-KUTTA TYPE METHOD FOR THE NUMERICAL SOLUTION OF INITIAL VALUE PROBLEM

Implicit Runge- Kutta methods are used for solving stiff problems which mostly arise in real life situations. Analysis of the order, error constant, consistency and convergence will help in determining an effective Runge- Kutta Method (RKM) to use. Due to the loss of linearity in Runge –Kutta Methods and the fact that the general Runge –Kutta Method makes no mention of the differential equation makes it impossible to define the order of the method independently of the differential equation.
 In this paper, we examine in simpler details how to obtain the order, error constant, consistency and convergence of a Runge -Kutta Type method (RKTM) when the step number .

A high-order implicit–explicit Runge–Kutta type scheme for the numerical solution of the Kuramoto–Sivashinsky equation

This manuscript is concerned with the development and the implementation of a numerical scheme to study the spatio-temporal solution profile of the well-known Kuramoto–Sivashinsky equation with appropriate initial and boundary conditions. A fourth-order Runge–Kutta based implicit–explicit scheme in time along with compact higher-order finite difference scheme in space is introduced. The proposed scheme takes full advantage of the method of line (MOL) and partial fraction decomposition techniques, therefore, it just needs to solve two backward Euler-type linear systems at each time step to get the solution. Performance of the scheme is investigated by testing it on some test examples and by comparing numerical results with relevant known schemes. The numerical results showed that the proposed scheme is more accurate and reliable than existing schemes to solve Kuramoto–Sivashinsky equation.

Global dynamical time for binary point-like particle systems

A geometrical construction for a global dynamical time for binary point-like particle systems, modeled by relativistic equations of motions, is presented. Thus, we provide a convenient tool for the calculation of the dynamics of recent models for the dynamics of black holes that use individual proper times. The construction is naturally based on the local Lorentzian geometry of the spacetime considered. Although in this presentation we are dealing with the Minkowskian spacetime, the construction is useful for gravitational models that have as a seed Minkowski spacetime. We present the discussion for a binary system, but the construction is obviously generalizable to multiple particle systems. The calculations are organized in terms of the order of the corresponding relativistic forces. In particular, we improve on the Darwin and Landau–Lifshitz approaches, for the case of electromagnetic systems. We discuss the possibility of a Lagrangian treatment of the retarded effects, depending on the nature of the relativistic forces. The higher-order contractions are based on a Runge–Kutta type procedure, which is used to calculate the quantities at the required retarded time, by increasing evaluations of the forces at intermediate times. We emphasize the difference between approximation orders in field equations and approximation orders in retarded effects in the equations of motion. We show this difference by applying our construction to the binary electromagnetic case.

НЕЛИНЕЙНАЯ ДИНАМИКА ТОПОЛОГИЧЕСКИ ОПТИМАЛЬНОЙ НАНО БАЛКИ ТИМОШЕНКО НА ОСНОВЕ МОДИФИЦИРОВАННОЙ МОМЕНТНОЙ ТЕОРИИ

Актуальность исследования. Для контроля данных технических операций бурения скважин, различного вида ремонта скважин, геологоразведочного бурения в нефтяной и газовой промышленности на буровых и ремонтных установках всех типов используются технологические датчики (датчики нагрузки, давления, температуры жидкости, выхода раствора, плотности жидкости). Современные датчики имеют малые размеры и для повышения чувствительности изготавливаются на базе наноэлектромеханических систем, которые включают в себя составляющие элементы, нанобалки и нанопластины. Эти элементы работают при высоких температурах и подвергаются механическим нагрузкам различного вида. Увеличение прочности этих элементов является несомненно актуальной задачей. Цель: построить математическую модель на базе кинематической гипотезы Тимошенко и модифицированной моментной теории чувствительного элемента в виде балки, нано электромеханического датчика под действием механических и тепловых полей;создать методологию получения оптимальной топологии нанобалки для произвольнойстатической и динамической нагрузкии различных граничных условий с целью увеличения ее жесткости;провести сравнительный анализ статики и нелинейной динамики оптимальной и неоптимальной балок. Объекты: элемент наноэлектромеханических систем в виде балки с учетом оптимальной микроструктуры. Методы: методы топологической оптимизации, вариационные методы, метод конечных разностей второго порядка, методытипа Рунге–Кутта, Фурье и вейвлет анализ, фазовый портрет и сечение Пуанкаре. Результаты. Построена методология получения оптимальной микроструктуры нанобалки на основе топологической оптимизации.На основе принципа Гамильтона–Остроградского построена математическаямодель неоднородной в двух направлениях (по толщине и длине) нанобалки Тимошенко на базе модифицированной моментной теории. Проведен сравнительный анализ статического изгиба и нелинейной динамики оптимальной и неоптимальной нанобалок.

On Two-Derivative Runge–Kutta Type Methods for Solving u‴ = f(x,u(x)) with Application to Thin Film Flow Problem

A class of explicit Runge–Kutta type methods with the involvement of fourth derivative, denoted as two-derivative Runge–Kutta type (TDRKT) methods, are proposed and investigated for solving a special class of third-order ordinary differential equations in the form u ‴ ( x ) = f ( x , u ( x ) ) . In this paper, two stages with algebraic order four and three stages with algebraic order five are presented. The derivation of TDRKT methods involves single third derivative and multiple evaluations of fourth derivative for every step. Stability property of the methods are analysed. Accuracy and efficiency of the new methods are exhibited through numerical experiments.