Explicit Runge-Kutta Methods Research Articles

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.

A linear, high-order, and unconditionally energy stable scheme for the epitaxial thin film growth model without slope selection

The epitaxial thin film growth model without slope selection is the L2-gradient flow of energy with a logarithmic potential in terms of the gradient of a height function. A challenge to numerically solving the model is how to treat the nonlinear term to preserve energy stability without compromising accuracy and efficiency. To resolve this problem, we present a high-order energy stable scheme by placing the linear and nonlinear terms in the convex and concave parts, respectively, and employing the specially designed implicit–explicit Runge–Kutta method. As a result, our scheme is linear, high-order accurate in time, and unconditionally energy stable. We show analytically that the scheme is unconditionally uniquely solvable and energy stable. Numerical experiments are presented to demonstrate the accuracy, efficiency, and energy stability of the proposed scheme.

CONSTRUCTION OF EXPLICIT RUNGE-KUTTA METHODS FOR MODELING OF DYNAMIC SYSTEMS WITH DELAY

A system of differential equations with delay is considered, which is a mathematical model of many technical processes with a time delay. Numerical Runge-Kutta methods and the method of expansion onTaylorseries on time delay are most often used to model such systems.When the value of the delay is greater than the step of numerical integration, the numerical solution of these systems is not difficult. For this, interpolation of the prehistory of the model and numerical methods for conventional systems of differential equations are used. For example, explicit Runge-Kutta methods. Using the method of steps, a numerical solution is obtained for the required period of time. But, if the time interval is large enough in comparison with the delay, then the number of steps turns out to be large, which slows down the process of numerical integration and leads to the accumulation of errors.For small delays,Taylortime delay expansions are used and the further solution of the usual system of differential equations by a numerical method, for example, by the Runge-Kutta method. This approach has limitations on the amount of delay and is not applicable for many models.Thus, it is often necessary to apply a step of numerical integration that is larger than the delay value and continuous implicit Runge-Kutta methods, which leads to a complication of the numerical algorithm, since at each step of numerical integration it is necessary to solve systems of nonlinear equations.In this paper, based on the construction of Newton's and Taylor's polynomials, an algorithm has been developed that allows using explicit Runge-Kutta methods to solve systems with delay and the step size of numerical integration is more than the delay value. For systems with delay, an explicit Runge-Kutta method of the fifth order of approximation is constructed on the basis of the most frequently used explicit Runge-Kutta methods of the fifth order of approximation. These methods are convenient in programming, have a higher speed of calculation than implicit methods, and are applicable for steps of numerical integration that are large in comparison with the lag.

HERK integration of finite-strain fully anisotropic plasticity models

For finite strain plasticity, we use the multiplicative decomposition of the deformation gradient to obtain a differential-algebraic system (DAE) in the semi-explicit form and solve it by a half-explicit algorithm. The terminology HERK is synonym of Half-Explicit Runge-Kutta method for DAE. The source is here the right Cauchy-Green tensor and an exact Jacobian of the second Piola-Kirchhoff stress is determined with respect to this source. The system is composed by a smooth nonlinear first-order differential equation and a non-smooth algebraic equation. The development of a half-explicit constitutive integrator is the content of this work. The integration makes use of an explicit Runge-Kutta method for the flow law complemented by the yield constraint. The flow law is a first-order differential equation and the yield constraint (including the loading/unloading conditions) is seen as the invariant of system. A half-explicit method is adopted to ensure satisfaction of the invariant. The resulting scalar equation is solved by the Newton-Raphson method to obtain the plastic multiplier. We make use of the elastic Mandel stress construction, which is power-consistent with the plastic strain rate. Iso-error maps are presented for a combination of Neo-Hookean material using the Hill yield criterion and a associative flow law. Two complete numerical examples are presented.

On the realization of explicit Runge-Kutta schemes preserving quadratic invariants of dynamical systems

We implement several explicit Runge-Kutta schemes that preserve quadratic invariants of autonomous dynamical systems in Sage. In this paper, we want to present our package ex.sage and the results of our numerical experiments. In the package, the functions rrk_solve, idt_solve and project_1 are constructed for the case when only one given quadratic invariant will be exactly preserved. The function phi_solve_1 allows us to preserve two specified quadratic invariants simultaneously. To solve the equations with respect to parameters determined by the conservation law we use the elimination technique based on Grbner basis implemented in Sage. An elliptic oscillator is used as a test example of the presented package. This dynamical system has two quadratic invariants. Numerical results of the comparing of standard explicit Runge-Kutta method RK(4,4) with rrk_solve are presented. In addition, for the functions rrk_solve and idt_solve, that preserve only one given invariant, we investigated the change of the second quadratic invariant of the elliptic oscillator. In conclusion, the drawbacks of using these schemes are discussed.

Explicit high-order energy-preserving methods for general Hamiltonian partial differential equations

A novel class of explicit high-order energy-preserving methods are proposed for general Hamiltonian partial differential equations with non-canonical structure matrix. When the energy is not quadratic, it is firstly done that the original system is reformulated into an equivalent form with a modified quadratic energy conservation law by the energy quadratization approach. Then the resulting system that satisfies the quadratic energy conservation law is discretized in time by combining explicit high-order Runge–Kutta methods with orthogonal projection techniques. The proposed schemes are shown to share the order of explicit Runge–Kutta method and thus can reach the desired high-order accuracy. Moreover, the methods are energy-preserving and explicit because the projection step can be solved explicitly. Numerical results are addressed to demonstrate the remarkable superiority of the proposed schemes in comparison with other structure-preserving methods.

A radial basis function finite difference (RBF-FD) method for numerical simulation of interaction of high and low frequency waves: Zakharov–Rubenchik equations

In this study, we examine numerical solutions of Zakharov–Rubenchik system which is a coupled nonlinear partial differential equation. The numerical method in the current study is based on radial basis function finite difference (RBF-FD) meshless method and an explicit Runge–Kutta method. As a radial basis function we choose polyharmonic spline augmented with polynomials. The essential motivation for choosing polyharmonic spline is that it is free of shape parameter which has a crucial role in accuracy and stability of meshless methods. The main benefit of the proposed method is the approximation of the differential operators is performed on local-support domain which produces sparse differentiation matrices. This reduces computational cost remarkably. To see performance of the proposed method, some test problems are solved. L∞ error norms and conserved quantities such as mass and energy are calculated. Numerical outcomes are presented and compared with other methods available in the literature. From the comparison it can be deduced that the proposed method gives reliable and precise results with low computational cost. Stability of the proposed method is also discussed.

Inverse Hybrid Linear Multistep Methods for Solving the Second Order Initial Value Problems in Ordinary Differential Equations

Inverse linear multistep methods (ILMMs) for first and second order differential equations have been proved to be suitable numerical methods for the solution of inverse initial value problems (IVPs). This paper presents the hybrid version of the ILMMs for the numerical solution of second order inverse IVPs. The stability of the proposed methods is represented in the boundary locus graph. The applicability of the schemes is demonstrated herein for the solution of linear and nonlinear problems. Computational results on the problems are compared with those from the existing method and ode45 (the explicit Runge–Kutta method).

Time-Accurate and highly-Stable Explicit operators for stiff differential equations

Unconditionally stable implicit time-marching methods are powerful in solving stiff differential equations efficiently. In this work, a novel framework to handle stiff physical terms implicitly is proposed. Both physical and numerical stiffness originating from convection, diffusion and source terms (typically related to reaction) can be handled by a set of predefined Time-Accurate and highly-Stable Explicit (TASE) operators in a unified framework. The proposed TASE operators act as preconditioners on the stiff terms and can be deployed to any existing explicit time-marching methods straightforwardly. The resulting time integration methods remain the original explicit time-marching schemes, yet with nearly unconditional stability. The TASE operators can be designed to be arbitrarily high-order accurate with Richardson extrapolation such that the accuracy order of original explicit time-marching method is preserved. Theoretical analyses and stability diagrams show that the s-stages sth-order explicit Runge-Kutta (RK) methods are unconditionally stable when preconditioned by the TASE operators with order p≤s and p≤2. On the other hand, the sth-order RK methods preconditioned by the TASE operators with order of p≤s and p>2 are nearly unconditionally stable. The only free parameter in TASE operators can be determined a priori based on stability arguments. Unlike classical implicit methods, the TASE methodology allows for solving non-linear problems with arbitrary order without requiring solving a nonlinear system of equations. A set of benchmark problems with strong stiffness is simulated to assess the performance of the TASE method. Numerical results suggest that the proposed framework preserves the high-order accuracy of the explicit time-marching methods with very-large time steps for all the considered cases. As an alternative to established implicit strategies, TASE method is promising for the efficient computation of stiff physical problems.

ESERK Methods to Numerically Solve Nonlinear Parabolic PDEs in Complex Geometries: Using Right Triangles

In this paper Extrapolated Stabilized Explicit Runge-Kutta methods (ESERK) are proposed to solve nonlinear partial differential equations (PDEs) in right triangles. These algorithms evaluate more times the function than a standard explicit Runge--Kutta scheme ($n_t$ times per step), and these extra evaluations do not increase the order of convergence but the stability region grows with $\mathcal{O}(n_t^2)$. Hence, the total computational cost is $\mathcal{O}(n_t)$ times lower than with a traditional explicit algorithm. Thus, these algorithms have been traditionally considered to solve stiff PDEs in squares/rectangles or cubes. In this paper, for the first time, ESERK methods are considered in a right triangle. It is demonstrated that such type of codes keep the convergence and the stability properties under certain conditions. This new approach would allow to solve nonlinear parabolic PDEs with stabilized explicit Runge--Kutta schemes in complex domains, that would be decomposed in rectangles and right triangles.

Efficient two-step Runge-Kutta methods for fluid dynamics simulations

Explicit two-step Runge-Kutta (TSRK) methods offer an efficient alternative to traditional explicit Low-Storage Runge-Kutta (LSRK) schemes for solving the Navier-Stokes equations. A special class of TSRK methods that reduce requirement compared to previous TSRK schemes are derived. Schemes of fourth, fifth and sixth order are implemented and tested.The new schemes are evaluated with two common test cases, a 2D cylinder and a 3D Taylor-Green vortex. The results are compared with classical time discretization strategies. Timings obtained in three different hardware configurations show that the new TSRK methods of order four are 25% faster than LSRK schemes of the same order.Fifth and sixth order TSRK methods are tested with the same 3D test case and the results are compared to LSRK algorithms. Results show TSRK schemes of fifth and sixth order are competitive compared to LSRK methods of the same orders, as LSRK methods are of second order for non linear differential systems.

One-step semi-implicit integration of general finite-strain plasticity models

Using the Kroner–Lee elastic and plastic decomposition of the deformation gradient, a differential-algebraic system is obtained (in the so-called semi-explicit form). The system is composed by a smooth nonlinear differential equation and a non-smooth algebraic equation. The development of an efficient one-step constitutive integrator is the goal of this work. The integration procedure makes use of an explicit Runge–Kutta method for the differential equation and a smooth replacement of the algebraic equation. The resulting scalar equation is solved by the Newton–Raphson method to obtain the plastic multiplier. We make use of the elastic Mandel stress construction, which is power-consistent with the plastic strain rate. Iso-error maps are presented for a combination of Neo-Hookean material using the Hill yield criterion and a associative flow law. A variation of the pressurized plate is presented. The exact Jacobian for the constitutive system is presented and the steps for use within a structural finite element formulation are described .

Non-linearly stable reduced-order models for incompressible flow with energy-conserving finite volume methods

A novel reduced-order model (ROM) formulation for incompressible flows is presented with the key property that it exhibits non-linearly stability, independent of the mesh (of the full order model), the time step, the viscosity, and the number of modes. The two essential elements to non-linear stability are: (1) first discretise the full order model, and then project the discretised equations, and (2) use spatial and temporal discretisation schemes for the full order model that are globally energy-conserving (in the limit of vanishing viscosity). For this purpose, as full order model a staggered-grid finite volume method in conjunction with an implicit Runge-Kutta method is employed. In addition, a constrained singular value decomposition is employed which enforces global momentum conservation. The resulting ‘velocity-only’ ROM is thus globally conserving mass, momentum and kinetic energy. For non-homogeneous boundary conditions, a (one-time) Poisson equation is solved that accounts for the boundary contribution. The stability of the proposed ROM is demonstrated in several test cases. Furthermore, it is shown that explicit Runge-Kutta methods can be used as a practical alternative to implicit time integration at a slight loss in energy conservation.

Highly efficient invariant-conserving explicit Runge-Kutta schemes for nonlinear Hamiltonian differential equations

A unified framework of invariant-conserving explicit Runge-Kutta schemes for the nonlinear Hamiltonian ODEs and PDEs are proposed by utilizing the invariant energy quadratization technique. First, the nonlinear Hamiltonian differential equation is transformed into an equivalent reformulation which admits a quadratic energy invariant. For the nonlinear ODE, the reformulation is then discretized using a class of relaxation Runge-Kutta schemes. For the nonlinear PDE, the reformulation is first discretized in the space direction by adopting the Fourier pseudo-spectral discretization which preserves the semi-discrete quadratic conservation laws. Then the semi-discrete system is integrated in the time direction using the explicit relaxation Runge-Kutta method. The obtained method can conserve different quadratic conservation laws to machine accuracy, which improves the numerical stability during long time computations. Besides, the proposed method keeps the same convergence rate of the standard Runge-Kutta scheme when computed with time step size rescaling, and gives a rate of convergence reduced by at most one when computed without step size rescaling. Numerical experiments for several ODEs and PDEs are provided to illustrate the advantages of the proposed algorithms over long time and verify the theoretical analysis.

Fully-Discrete Analysis of High-Order Spatial Discretizations with Optimal Explicit Runge–Kutta Methods

High-order unstructured methods have become a popular choice for the simulation of complex unsteady flows. Flux reconstruction (FR) is a high-order spatial discretization method, which has been found to be particularly accurate for scale-resolving simulations of complex phenomena. In addition, it has been shown to provide sufficient dissipation for implicit large-eddy simulation (ILES). In conjunction with an FR discretization, an appropriate temporal scheme must be chosen. A common choice is explicit schemes due to their efficiency and ease of implementation. However, these methods usually require a small time-step size to remain stable. Recently, the development of optimal explicit Runge–Kutta (OERK) schemes has enabled stable simulations with larger time-step sizes. Hence, we analyze the fully-discrete properties of the FR method with OERK temporal schemes. We show results for first, second, third, fourth and eighth-order OERK schemes. We observe that OERK schemes modify the spectral behaviour of the semidiscretization. In particular, dissipation decreases in the region of high wavenumbers. We observe that higher-order OERK schemes require a smaller time step than the low-order schemes. However, they follow the dispersion relations of the FR scheme for a larger range of wavenumbers. We validate our analysis with simple advection test cases. It was observed that first and second-degree temporal schemes introduce a relatively large amount of error in the solutions. A one-dimensional ILES test case showed that, as long as the time-step size is not in the vicinity of the stability limit, results are generally similar to classical RK schemes.

On strong stability of explicit Runge–Kutta methods for nonlinear semibounded operators

AbstractExplicit Runge–Kutta methods are classical and widespread techniques in the numerical solution of ordinary differential equations (ODEs). Considering partial differential equations, spatial semidiscretizations can be used to obtain systems of ODEs that are solved subsequently, resulting in fully discrete schemes. However, certain stability investigations of high-order methods for hyperbolic conservation laws are often conducted only for the semidiscrete versions. Here, strong stability (also known as monotonicity) of explicit Runge–Kutta methods for ODEs with nonlinear and semibounded (also known as dissipative) operators is investigated. Contrary to the linear case it is proven that many strong-stability-preserving (SSP) schemes of order 2 or greater are not strongly stable for general smooth and semibounded nonlinear operators. Additionally, it is shown that there are first-order-accurate explicit SSP Runge–Kutta methods that are strongly stable (monotone) for semibounded (dissipative) and Lipschitz continuous operators.

Numerical Simulation of Micropolar Flow in a Channel under Osciatory Pressure Gradient

We numerically investigate the pulsatile flow and heat transfer of a micropolar fluid through a Darcy-Forchhmeir porous channel in the presence of wall transpiration. We use the central difference approximations for the spatial derivatives, whereas the time integration has been performed by employing the three steps explicit Runge-Kutta method to obtain the numerical solution. It is noted that the Darcy parameter tends to accelerate the fluid, whereas the Forchheimer quadratic drag parameter and the magnetic parameter would reduce the flow velocity. The effect of the steady component of the pressure gradient is to remarkably accelerate the flow whereas that of the oscillatory component is time-dependent. An increase in the Prandtl number tends to almost straighten the temperature profiles.

Dynamic analysis of a hyper-redundant space manipulator with a complex rope network

This paper studies the dynamics of a space manipulator. The space manipulator is designed for precise on-orbit servicing missions in a highly constrained environment. The concerned manipulator has hyper-redundant degrees of freedom and moves with a piecewise constant curvature, which enhances the flexibility and controllability. Such manipulator consists of a large number of links and a complex rope network. When the manipulator is driven, the interacting forces between the links and ropes introduce complexity into the dynamic behavior. In terms of dynamic modeling, the manipulator is a very complex system. This paper proposes a dynamic model of the manipulator based on methods of multibody dynamics. The ropes are assumed to be massless and linear elastic. The equations of motion are derived using space operator algebra. The vibration of the manipulator is investigated. The governing equations of the vibration are derived by applying the perturbation method to the proposed dynamic model. The values of the natural frequencies are investigated for the elasticities of the ropes. The proposed dynamic model is also applied in numerical simulation. The explicit fourth-order Runge-Kutta method is utilized to solve the equations of motion numerically. In numerical simulation, an upper bound of the time step is encountered. The value of upper bound is found to be related to the elasticities of the ropes. Such phenomena are studied by analyzing the stability region of the Runge-Kutta methods. Besides, the computational efficiency of the numerical simulation is also limited by the value of upper bound. Two modifications of the dynamic model are introduced to relax the upper bound of the time step. The effects of the modifications are demonstrated by numerical results.

Solving semi-linear stiff neutral equations by implicit–explicit Runge-Kutta methods

In this paper, for solving semi-linear stiff neutral equations, a class of extended implicit–explicit Runge-Kutta (EIERK) methods are suggested. Under some suitable conditions, it is shown that an EIERK method is convergent of order one or two. The presented numerical examples further verify the theoretical results and effectiveness of the methods. A comparison with the adapted fully implicit Runge-Kutta methods shows that EIERK methods possess the higher computational efficiency.

A 2-Stage Implicit Runge-Kutta Method Based on Heronian Mean for Solving Ordinary Differential Equations

In recent times, the use of different types of mean in the derivation of explicit Runge-Kutta methods had been on increase. Researchers have explored explicit Runge-Kutta methods derivation by using different types of mean such as geometric mean, harmonic mean, contra-harmonic mean, heronian mean to name but a few; as against the conventional explicit Runge-Kutta methods which was viewed as arithmetic mean. However, despite efforts to improve the derivation of explicit Runge-Kutta methods with use of other types of mean, none has deemed it fit to extend this notion to implicit Runge-Kutta methods. In this article, we present the use of heronian mean as a basis for the construction of implicit Runge-Kutta method in a way of improving the conventional method which is arithmetic mean based. Numerical results was conducted on ordinary differential equations which was compared with the conventional two-stage fourth order implicit Runge-Kutta (IRK4) method and two-stage third order diagonally implicit Runge-Kutta (DIRK3) method. The results presented confirmed that the new scheme performs better than these numerical methods. A better Qualitative properties using Dalquist test equation were established.