Numerical experiments with some explicit pseudo two-step RK methods on a shared memory computer

In this paper we study a class of explicit pseudo two-step Runge-Kutta (EPTRK) methods for first-order ODEs for parallel computers. We investigate linear stability and derive methods with enlarged stability regions. In numerical experiments on a shared memory computer we compare a parallel variable step size EPTRK implementation with the effcient sequential Runge-Kutta method dopri5.

The standard approach to applying IRK methods in the solution of two-point boundary value problems involves the solution of a non-linear system ofn×s equations in order to calculate the stages of the method, wheren is the number of differential equations ands is the number of stages of the implicit Runge-Kutta method. For two-point boundary value problems, we can select a subset of the implicit Runge-Kutta methods that do not require us to solve a non-linear system; the calculation of the stages can be done explicitly, as is the case for explicit Runge-Kutta methods. However, these methods have better stability properties than the explicit Runge-Kutta methods. We have called these new formulas two-point explicit Runge-Kutta (TPERK) methods. Their most important property is that, because their stages can be computed explicity, the solution of a two-point boundary value problem can be computed more efficiently than is possible using an implicit Runge-Kutta method. We have also developed a symmetric subclass of the TPERK methods, called ATPERK methods, which exhibit a number of useful properties.

Variational formulations for explicit Runge-Kutta Methods

This paper is aimed at analyzing the consistency and convergence of a derived explicit fourth-stage fourth-order Runge-Kutta method. The analysis revealed that the method is consistent and convergent. The implementation of this method on initial-value problems was done in previous paper, and it revealed that the method compared favorably well with the existing classical fourth stage fourth order explicit Runge Kutta method.

Explicit Runge–Kutta methods and implicit multistep methods utilizing a Newton–Krylov nonlinear solver are evaluated for a range of configurations of the shallow-water dynamical core of the spectral element community atmosphere model to evaluate their computational performance. These configurations are designed to explore the attributes of each method under different but relevant model usage scenarios including varied spectral order within an element, static regional refinement, and scaling to the largest problem sizes. This analysis is performed within the shallow-water dynamical core option of a full climate model code base to enable a wealth of simulations for study, with the aim of informing solver development within the more complete hydrostatic dynamical core used for climate research. The limitations and benefits to using explicit versus implicit methods, with different parameters and settings, are discussed in light of the trade-offs with Message Passing Interface (MPI) communication and memory and their inherent efficiency bottlenecks. Given the performance behavior across the configurations analyzed here, the recommendation for future work using the implicit solvers is conditional based on scale separation and the stiffness of the problem. For the regionally refined configurations, the implicit method has about the same efficiency as the explicit method, without considering efficiency gains from a preconditioner. The potential for improvement using a preconditioner is greatest for higher spectral order configurations, where more work is shifted to the linear solver. Initial simulations with OpenACC directives to utilize a Graphics Processing Unit (GPU) when performing function evaluations show improvements locally, and that overall gains are possible with adjustments to data exchanges.

In this paper, we analyze any-order Runge-Kutta spectral volume schemes (RKSV(s,k)) for solving the one-dimensional scalar hyperbolic equation. The RKSV(s,k) was constructed by using the $s$-th explicit Runge-Kutta method in time-discretization which has strong-stabilitypreserving (SSP) property, and by letting a piecewise $k$−th degree($k$ ≥ 1 is an arbitrary integer) polynomial satisfy the local conservation law in each control volume designed by subdividing the underlying mesh with $k$ Gauss-Legendre points (LSV) or right-Radau points (RRSV). For the RKSV(s,k), we would like to establish a general framework which use the matrix transferring process technique for analyzing the stability and the convergence property. The framework for stability is evolved based on the energy equation, while the framework for error estimate is evolved based on the error equation. And the evolution process is represented by matrices. After the evolution is completed, three key indicative pieces of information are obtained: the termination factor $ζ$, the indicator factor $ρ$, and the final evolved matrix. We prove that for the RKSV(s,k), the stability holds and the $L_2$ norm error estimate is $\mathcal{O}$($h^{k+1}$ + $τ^s$ ), provided that the CFL condition is satisfied. Our theoretical findings have been justified by several numerical experiments.

This paper investigates a class of explicit pseudo three-step Runge-Kutta methods for arbitrarily high order nonstiff initial value problems for systems of first-order differential equations. By using collocation techniques and by suitably choosing collocation points we can obtain a stable \(s\)-stage explicit pseudo three-step Runge-Kutta method (EPThRK method) of order \(p=2s\) requiring only one effective sequential f-evaluation per step on \(s\)-processor parallel computers. By a few widely-used test problems, we show the superiority of the new EPThRK methods proposed in this paper over red well-known parallel PIRK codes and efficient sequential ODEX, DOPRI5 and DOP853 codes available in the literature.

A trigonometrically fitted explicit Numerov-type method for second-order initial value problems with oscillating solutions

We propose a unified theoretical framework to examine the energy dissipation properties at all stages of explicit exponential Runge-Kutta (EERK) methods for gradient flow problems. The main part of the novel framework is to construct the differential form of EERK method by using the difference coefficients of method and the so-called discrete orthogonal convolution kernels. As the main result, we prove that an EERK method can preserve the original energy dissipation law unconditionally if the associated differentiation matrix is positive semi-definite. A simple indicator, namely average dissipation rate, is also introduced for these multi-stage methods to evaluate the overall energy dissipation rate of an EERK method such that one can choose proper parameters in some parameterized EERK methods or compare different kinds of EERK methods. Some existing EERK methods in the literature are evaluated from the perspective of preserving the original energy dissipation law and the energy dissipation rate. Some numerical examples are also included to support our theory.

Explicit Runge-Kutta pairs are known to provide efficient solutions to initial value differential equations with inexpensive derivative evaluations. Two-step Runge-Kutta methods strive to improve the efficiency by utilizing approximations to the solution and its derivatives from the previous step. This article suggests a strategy for computing embeddedpairs of such two-step methods using a smaller number of function evaluations than that required for traditional Runge-Kutta methods of the same order. This leads to the efficient and reliable estimation of local discretization error and a robust step control strategy. The change of stepsize is achieved by a suitable interpolation of stage values from the previous step and does not require any additional function evaluations. Two examples illustrate the features of these pairs.

The stability region of an explicit and consistentm-stage Runge-Kutta method cannot contain the closed disk with diameter [−2m, 0] as a proper subset.

On the economization of explicit Runge-Kutta methods

High-order linearly implicit two-step peer – finite element methods for time-dependent PDEs

The diffusion equation is discretized in spacial direction and transformed into the ordinary differential equations. The ordinary differential equations are solved by Lie group method and the explicit Runge-Kutta method. Numerical results showed that Lie group method is more stable than the corresponding explicit Runge-Kutta method.

In this paper, an application of symplectic integrators for solving an important problem of non-linear dynamics is considered. Numerical algorithms for solving six-body problem are described, built on explicit second order Runge-Kutta method, implicit Euler method, symplectic Verlet method and D-method. The examined six-body problem time-domain solution is given, as well as the system's full energy dynamics for the implemented numerical algorithm. Conclusions of symplectic integrators efficiency are made based on simulation results.