Explicit Runge-Kutta Methods Research Articles

A Method for Choosing the Spatial and Temporal Approximations for the LES Approach

Analysis and optimization of the hybrid upwind-central numerical methods for modern versions of large eddy simulations (LESs) are presented herein. Optimization was performed based on examination of the characteristics of the spatial and temporal finite-volume approximations of the convective terms of filtered Navier–Stokes equations. A method for selecting level of subgrid viscosity that corresponds to the chosen numerical scheme and makes it possible to obtain an extended inertial interval of the energy spectrum is proposed. A series of LESs of homogeneous isotropic turbulence decay were carried out, and the optimal values of the subgrid model constants included in the hybrid shear stress transport delay detached eddy simulation (SST-DDES) method were determined. A procedure for generating a consistent initial field of subgrid parameters for these simulations is described. The three-stage explicit Runge–Kutta method was demonstrated to be sufficient for stable time integration, while the popular explicit midpoint method was not. The slope of the energy spectrum was shown to be almost independent of the central-difference scheme order and of the mesh spacing when the correct numerical method was applied. Numerical viscosity was found to become much greater than subgrid viscosity when the upwind scheme contributed about 10% or more to the convective flux approximation.

Energy-conserving explicit and implicit time integration methods for the multi-dimensional Hermite-DG discretization of the Vlasov-Maxwell equations

We study the conservation properties of the Hermite-discontinuous Galerkin (Hermite-DG) approximation of the Vlasov-Maxwell equations. In this semi-discrete formulation, the total mass is preserved independently for every plasma species. Further, an energy invariant exists if central numerical fluxes are used in the DG approximation of Maxwell's equations, while a dissipative term is present when upwind fluxes are employed. In general, traditional temporal integrators might fail to preserve invariants associated with conservation laws during the time evolution. Hence, we analyze the capability of explicit and implicit Runge-Kutta (RK) temporal integrators to preserve such invariants. Since explicit RK methods can only ensure preservation of linear invariants but do not provide any control on the system energy, we consider modified explicit RK methods in the family of relaxation Runge-Kutta methods (RRK). These methods can be tuned to preserve the energy invariant at the continuous or semi-discrete level, a distinction that is important when upwind fluxes are used in the discretization of Maxwell's equations since upwind provides a numerical source of energy dissipation that is not present when central fluxes are used. We prove that the proposed methods are able to preserve the energy invariant and to maintain the semi-discrete energy dissipation (if present) according to the discretization of Maxwell's equations. An extensive set of numerical experiments corroborates the theoretical findings. It also suggests that maintaining the semi-discrete energy dissipation when upwind fluxes are used leads to an overall better accuracy of the method relative to using upwind fluxes while forcing exact energy conservation.

Explicit pseudo-symplectic Runge-Kutta methods for stochastic Hamiltonian systems

We give conditions for stochastic Runge-Kutta methods to near preserve quadratic invariants, and we discuss the associated simplified order conditions. For stochastic Hamiltonian systems we propose a systematic approach to construct explicit stochastic Runge-Kutta pseudo-symplectic schemes. Our approach is based on colored trees and B-series. We construct some pseudo-symplectic stochastic Runge-Kutta methods with strong convergence order, and we illustrate numerically the long term performance of the proposed schemes.

COSEν: A collective oscillation simulation engine for neutrinos

We introduce the implementation details of the simulation code COSEν, which numerically solves a set of non-linear partial differential equations that govern the dynamics of neutrino collective flavor conversions. We systematically provide the details of both finite difference method supported by Kreiss-Oliger dissipation and finite volume method with seventh order weighted essentially non-oscillatory scheme. To ensure the reliability of the code, we perform comparison of the simulation results with theoretically obtainable solutions. In order to understand and characterize the error accumulation behavior of the implementations when neutrino self-interactions are switched on, we also analyze the evolution of the deviation of the conserved quantities for different values of simulation parameters. We report the performance of our code with both CPUs and GPUs. The public version of the COSEν package is available at https://github.com/COSEnu/COSEnu. Program summaryProgram Title:COSEνCPC Library link to program files:https://doi.org/10.17632/d4dhg2sths.1Developer's repository link:https://github.com/COSEnu/COSEnuCode Ocean capsule:https://codeocean.com/capsule/7929908Licensing provisions: BSD 3-clauseProgramming language:C++, Python 3.0+Supplementary material:https://cosenu.github.io/cosenu_docs/Nature of problem: Collective neutrino flavor oscillations is a novel but generic phenomenon that occurs in cosmological and astrophysical environments dense in neutrinos. Understanding this phenomenon requires solving the corresponding quantum kinetic transport equation, which is a set of nonlinear and hyperbolic partial differential equations.Solution method: We adopt the method of lines to solve the hyperbolic partial differential equations in our program. The spatial derivatives are evaluated using either the finite difference method supported by 3rd order Kreiss-Oliger dissipation or the finite volume method with 7th order WENO scheme. The time integration is performed using the explicit 4th order Runge-Kutta method. Parallelization with OpenMP or OpenACC is supported.

Numerical methods, energy conservation, and a new method for particle motion in magnetic fields

This article mathematically examines the energy conservation of explicit Runge–Kutta methods and basic symplectic integration methods for a simple harmonic oscillator and a particle in a magnetic field. With the demonstration of the failure of these methods to conserve the energy of a particle in a magnetic field, a new integration method is presented for this type of system where energy conservation is dictated by the type of floating point variable used, not the size of the time step or the method order. Higher order versions of the method improve the accuracy in the calculation of the particle position and allow for more accurate calculations of the velocity in a spatially varying field. Furthermore, the new method is demonstrated to have improved phase accuracy when compared to several common implicit methods.

On Energy Laws and Stability of Runge--Kutta Methods for Linear Seminegative Problems

This paper presents a systematic theoretical framework to derive the energy identities of general implicit and explicit Runge--Kutta (RK) methods for linear seminegative systems. It generalizes the stability analysis of explicit RK methods in [Z. Sun and C.-W. Shu, SIAM J. Numer. Anal., 57 (2019), pp. 1158-1182]. The established energy identities provide a precise characterization on whether and how the energy dissipates in the RK discretization, thereby leading to weak and strong stability criteria of RK methods. Furthermore, we discover a unified energy identity for all the diagonal Pade approximations, based on an analytical Cholesky type decomposition of a class of symmetric matrices. The structure of the matrices is very complicated, rendering the discovery of the unified energy identity and the proof of the decomposition highly challenging. Our proofs involve the construction of technical combinatorial identities and novel techniques from the theory of hypergeometric series. Our framework is motivated by a discrete analogue of integration by parts technique and a series expansion of the continuous energy law. In some special cases, our analyses establish a close connection between the continuous and discrete energy laws, enhancing our understanding of their intrinsic mechanisms. Several specific examples of implicit methods are given to illustrate the discrete energy laws. A few numerical examples further confirm the theoretical properties.

An Eulerian-Lagrangian Runge-Kutta finite volume (EL-RK-FV) method for solving convection and convection-diffusion equations

We propose a new Eulerian-Lagrangian Runge-Kutta finite volume method for numerically solving convection and convection-diffusion equations. Eulerian-Lagrangian and semi-Lagrangian methods have grown in popularity mostly due to their ability to allow large time steps. Our proposed scheme is formulated by integrating the PDE on a space-time region partitioned by approximations of the characteristics determined from the Rankine-Hugoniot jump condition; and then rewriting the time-integral form into a time differential form to allow application of Runge-Kutta (RK) methods via the method-of-lines approach. The scheme can be viewed as a generalization of the standard Runge-Kutta finite volume (RK-FV) scheme for which the space-time region is partitioned by approximate characteristics with zero velocity. The high-order spatial reconstruction is achieved using the recently developed weighted essentially non-oscillatory schemes with adaptive order (WENO-AO); and the high-order temporal accuracy is achieved by explicit RK methods for convection equations and implicit-explicit (IMEX) RK methods for convection-diffusion equations. Our algorithm extends to higher dimensions via dimensional splitting. Numerical experiments demonstrate our algorithm's robustness, high-order accuracy, and ability to handle extra large time steps.

ON THE CONSISTENCY AND CONVERGENCE ANALYSIS OF A DERIVED EXPLICIT FOURTH-STAGE FOURTH-ORDER RUNGE-KUTTA METHOD

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.

PARAMETRIC VARIATION OF AN EXPLICIT FOURTH-STAGE FOURTH-ORDER RUNGE-KUTTA METHOD WITH ABSOLUTE STABILITY

The purpose of this paper is to separate the f(y) functional derivatives by discarding all functional derivatives of f(x,y) after using Taylor Series expansion to expand the fourth-stage fourth-order explicit Runge-Kutta method so as to derive a reduced number of equations for easy computation. Efforts will be made to vary the parameters with the aim of getting a new explicit fourth-order formula that can improve results when implemented on initial-value problems. Efforts will also be made to carry out stability, convergence and consistency analysis and represent the derived equations, their individual f(y) functional derivatives and their various elementary differentials on Butcher’s rooted trees. This idea is derivable from general graphs and combinatorics.

Artificial intelligence knacks-based computing for stochastic COVID-19 SIRC epidemic model with time delay

Time delays play an important part in modeling the fact that one cannot be communicable for a long time after becoming sick. Delay can be triggered by a variety of epidemiological situations. The most egregious causes of a delay are infection latency in the vector and infection latency in the infected host. The dynamics of susceptible, infected, recovered and cross-immune (SIRC) classed-based model having cross-immune and time-delay in the transmission for spread of COVID-19 abbreviated as (SIRC-CTC-19) are investigated in this study using an intelligent numerical computing paradigm based on the Levenberg–Marquardt Method backpropagated by neural networks (LM-BPNN). The model is mathematically governed by a system of ordinary differential equations that depicts the four nodes as susceptible, infected, recovered and cross-immune ones (SIRC) nodes with cross-immune class and time-delay in transmission components for COVID-19 dissemination (CTC-19). The reference solution of the SIRC model for the spread of COVID-19 is produced by using the explicit Runge–Kutta method for the many scenarios of this model arising from altering delay with regard to time. This reference solution permits the use of evolutionary computing to solve the SIRC-CTC-19 using train, validate and test techniques. The proposed LM-BPNN method’s accuracy has been proven by its results overlapping with explicit Runge–Kutta results Calculation of regression metrics, error analysis of histogram illustrations and learning curves on MSE effectively augment the LM-BPNN’s accuracy, convergence and reliability in solving the SIRC-CTC-19 model.

Discrete LQR and ILQR methods based on high order Runge-Kutta discretizations

In this paper, discrete linear quadratic regulator (DLQR) and iterative linear quadratic regulator (ILQR) methods based on high-order Runge-Kutta (RK) discretizations are proposed for solving linear and nonlinear quadratic optimal control problems respectively. As discovered in Hager (2000) [11], direct approaches with RK discretization are equivalent with indirect approaches based on symplectic partitioned Runge-Kutta (SPRK) integration. In this paper, we will reconstruct this equivalence by the analogue of continuous and discrete dynamic programming. Then, based on the equivalence, we discuss the issue that the internal-stage controls produced by direct approaches may have lower order accuracy than the RK method used. We propose order conditions for internal-stage controls and then demonstrate that third or fourth order explicit RK methods cannot avoid the order reduction phenomenon. To overcome this obstacle, we calculate node control instead of internal-stage controls in DLQR and ILQR methods. Another advantage of our methods is high computational efficiency which comes from the usage of feedback technique. A simple numerical example will illustrate the validity of ILQR in solving nonlinear optimal control problems with high-order RK discretizations. In this paper, we also demonstrate that ILQR is essentially a quasi-Newton method with linear convergence rate.

Third-order Paired Explicit Runge-Kutta schemes for stiff systems of equations

The ability to advance locally-stiff systems of equations in time depends on accurate and efficient temporal schemes. Recently, a new family of Paired Explicit Runge-Kutta (P-ERK) methods has been proposed. This approach allows different Runge-Kutta schemes with different numbers of active stages to be assigned based on local stiffness criteria. Whereas the original P-ERK formulation was only second-order accurate, in this paper we propose a new third-order family. We present the general formulation for these schemes, and then optimize them for the high-order discontinuous Galerkin method recovered via the flux reconstruction approach. We then verify that these schemes obtain their designed third-order accuracy for non-linear systems of equations. Performance results show that they achieve speedup factors up to four relative to a classical third-order Runge-Kutta methods for laminar and turbulent flow over an SD7003 airfoil, and turbulent flow over a tandem sphere configuration. Based on these results, this new family of third-order P-ERK schemes provides an appealing approach for accurate and efficient solution of locally-stiff systems of equations.

Conservation Properties of Iterative Methods for Implicit Discretizations of Conservation Laws

Conservation properties of iterative methods applied to implicit finite volume discretizations of nonlinear conservation laws are analyzed. It is shown that any consistent multistep or Runge-Kutta method is globally conservative. Further, it is shown that Newton’s method, Krylov subspace methods and pseudo-time iterations are globally conservative while the Jacobi and Gauss-Seidel methods are not in general. If pseudo-time iterations using an explicit Runge-Kutta method are applied to a locally conservative discretization, then the resulting scheme is also locally conservative. However, the corresponding numerical flux can be inconsistent with the conservation law. We prove an extension of the Lax-Wendroff theorem, which reveals that numerical solutions based on these methods converge to weak solutions of a modified conservation law where the flux function is multiplied by a particular constant. This constant depends on the choice of Runge-Kutta method but is independent of both the conservation law and the discretization. Consistency is maintained by ensuring that this constant equals unity and a strategy for achieving this is presented. Simulations show that this strategy improves the convergence rate of the pseudo-time iterations. Experiments with GMRES suggest that it also suffers from inconsistency but that this is automatically accounted for after some finite number of iterations. Similar experiments with coarse grid corrections based on agglomeration indicate no inconsistency.

L-stable and A-stable numerical method of order two for stiff differential equation

This article presents a finite difference method of order two for stiff differential equation that will overcome the effects of stiffness since it is A-stable. And, reflect the asymptotic behavior of the solution of the stiff problem since it is L-stable. The classical explicit Runge–Kutta methods of order two are not suitable for stiff problems since the numerical solution will not overcome the effects of stiffness and will not reflect the asymptotic behavior of the solution of the stiff problem (for example, Dahlquist problem). And, they are not both A-stable and L-stable. On applying Euler’s implicit method, the numerical solution will overcome the effects of stiffness and will reflect the asymptotic behavior of the solution of the stiff problem. And hence, it is both A-stable and L-stable. But it is of order one method. A new numerical method which is both A-stable and L-stable and of order two for the numerical solution of a stiff differential equation is presented in this article. It will overcome the effects of stiffness and reflects the asymptotic behavior of the solution. The new method works well for large values of step size and works well in large domain. The new method is a modified form of the mid-point rule for the stiff problems. The rate of order of convergence is proved to be two both theoretically and numerically. Experimental results show the performance of the method based on the metrics such as stability function, stability region, order star fingers, the rate of order of theoretical and numerical convergence, absolute and relative errors, percentage of numerical solution accuracy and local and global truncation errors both numerically and graphically.

On the stability of explicit finite difference methods for advection–diffusion equations

AbstractIn this article we study the stability of explicit finite difference discretization of advection–diffusion equations (ADE) with arbitrary order of accuracy in the context of method of lines. The analysis first focuses on the stability of the system of ordinary differential equations that is obtained by discretizing the ADE in space and then extends to fully discretized methods in combination with explicit Runge–Kutta methods. In particular, we prove that all stable semi‐discretization of the ADE leads to a conditionally stable fully discretized method as long as the time‐integrator is at least first‐order accurate, whereas high‐order spatial discretization of the advection equation cannot yield a stable method if the temporal order is too low. In the second half of the article, the analysis and the stability results are extended to a partially dissipative wave system, which serves as a model for common practice in many fluid mechanics applications that incorporate a viscous stress in the momentum equation but no heat dissipation in the energy equation. Finally, the major theoretical predictions are verified by numerical examples.

A COMPARISON OF EXPLICIT RUNGE–KUTTA METHODS

AbstractRecent higher-order explicit Runge–Kutta methods are compared with the classic fourth-order (RK4) method in long-term integration of both energy-conserving and lossy systems. By comparing quantity of function evaluations against accuracy for systems with and without known solutions, optimal methods are proposed. For a conservative system, we consider positional accuracy for Newtonian systems of two or three bodies and total angular momentum for a simplified Solar System model, over moderate astronomical timescales (tens of millions of years). For a nonconservative system, we investigate a relativistic two-body problem with gravitational wave emission. We find that methods of tenth and twelfth order consistently outperform lower-order methods for the systems considered here.

Application of the lumped-parameter method for modeling nonlinear vibrations of drill strings with stabilizers in a supersonic gas flow

In this paper, we study the nonlinear dynamics of a drill string under the effect of external loads in a supersonic gas flow, taking into account supporting stabilizers. The case of the drill string spatial lateral vibrations, which are the main cause of the drilling equipment failures, is considered. To account for nonlinear effects, a mathematical model based on the general relations of Novozhilov's nonlinear theory of elasticity is developed. The discretization of the model is conducted using the lumped-parameter method. The explicit fourth order Runge-Kutta method is utilized for obtaining a numerical solution for the studied system. The effect of damping elements (supporting stabilizers) on the drill string lateral vibrations is analyzed. Numerical illustrations are presented, and the corresponding recommendations for deep and shallow drilling are given.

On the Choice of Methods for Numerically Integrating the Equations of Transients in Electric Power Systems

Algorithms for solving the system of differential-algebraic equations describing electromechanical transients in electric power systems are considered along with matters concerned with ensuring the reliability and required accuracy of the solution results. Classical explicit methods, methods for implicit numerical integration and simultaneous solution of differential-algebraic systems of equations are used to model dynamic processes in power systems. On the basis of the methods used, difference models of electric power system components are obtained. The equations of transients in electric power systems are written in a homogeneous coordinate basis with the use of nodal voltage equations. A comparative analysis of algorithms based on the explicit Runge-Kutta method and implicit methods of Euler, trapezoids, Euler with fitting coefficients is carried out. The analysis results have shown that in terms of stability and accuracy, the trapezoid method has the greatest advantages, which is not inferior to the 4th order Runge-Kutta method and allows calculations with a large integration step to be carried out. The effectiveness of the combined method is shown, in which the Euler method with fitting is used to solve the differential equations describing electromagnetic processes, and the trapezoid method is used to solve the equations of electromechanical motion.

Accelerating the numerical solution of thermal runaway in Li-ion batteries

Computational modeling and simulation of thermal runaway in Li-ion batteries is an effective tool to understand safety concerns associated with Li-ion batteries. However, thermal runaway models are often computationally expensive due to the highly non-linear and coupled nature of the equations involved. Therefore, accelerating the numerical computation of thermal runaway in Li-ion batteries while maintaining computational accuracy and stability is of utmost practical importance. This paper compares several time integration schemes to solve the highly non-linear and stiff differential equations that govern thermal runaway in Li-ion batteries. The schemes are implemented to solve a lumped capacitance thermal runaway model for a representative thermal abuse scenario. Results showed that explicit methods produce similar accuracy as implicit methods at a fraction of the computational cost while maintaining stability even for stiff thermal runaway scenarios. For example, a one-stage and a two-stage explicit Runge-Kutta method resulted in two orders of magnitude lower computational time than the implicit methods while maintaining a similar level of accuracy. By investigating the computational performance of various numerical schemes for analyzing thermal runaway, this work contributes towards improved safety of energy conversion and storage in Li-ion batteries.

A mathematical model for the corneal transparency problem

AbstractUnderstanding the physical basis of corneal transparency has been a subject of interest amongst physicists, basic scientists and ophthalmologists. Impairment of corneal clarity is a significant cause of visual morbidity worldwide. Thus, it is essential to understand the mechanisms behind corneal transparency and how the alterations due to corneal pathologies affect vision. We use Maxwell’s equations to model light propagation in ocular tissues and a nodal discontinuous Galerkin method combined with an explicit Runge-Kutta method to simulate light propagation in normal and pathological corneas. Our simulation results illustrate that an increase in the diameter of some fibres causes an increase in backscattering. Thus, these may represent some of the physical changes in the cornea that might result in loss of transparency and visual morbidity.