Stability Analysis and Performance Evaluation of Additive Mixed-Precision Runge-Kutta Methods

Additive Runge-Kutta methods designed for preserving highly accurate solutions in mixed-precision computation were previously proposed and analyzed.These specially designed methods use reduced precision for the implicit computations and full precision for the explicit computations.In this work,we analyze the stability properties of these methods and their sensitivity to the low-precision rounding errors,and demonstrate their performance in terms of accuracy and efficiency.We develop codes in FORTRAN and Julia to solve nonlinear systems of ODEs and PDEs using the mixed-precision additive Runge-Kutta(MP-ARK)methods.The convergence,accuracy,and runtime of these methods are explored.We show that for a given level of accuracy,suitably chosen MP-ARK methods may provide significant reductions in runtime.

Additive Runge-Kutta methods for H_2/O_2/Ar detonation with a detailed elementary chemical reaction model

We report here the additive Runge-Kutta methods for computing reactive Euler equations with a stiff source term, and in particular, their applications in gaseous detonation simulations. The source term in gaseous detonation is stiff due to the presence of wide range of time scales during thermal-chemical non-equilibrium reactive processes and some of these time scales are much smaller than that of hydrodynamic flow. The high order, L-stable, additive Runge-Kutta methods proposed in this paper resolved the stiff source term into the stiff part and non-stiff part, in which the stiff part was solved implicitly while the non-stiff part was handled explicitly. The proposed method was successfully applied to simulating the gaseous detonation in a stoichiometric H2/O2 /Ar mixture based on a detailed elementary chemical reaction model comprised of 9 species and 19 elementary reactions. The results showed that the stiffly accurate additive Runge-Kutta methods can capture the discontinuity well, and describe the detonation complex wave configurations accurately such as the triple wave structure and cellular pattern.

Stability and Time-Step Constraints of Implicit-Explicit Runge-Kutta Methods for the Linearized Korteweg-de Vries Equation

This paper provides a study on the stability and time-step constraints of solving the linearized Korteweg-de Vries(KdV)equation,using implicit-explicit(IMEX)Runge-Kutta(RK)time integration methods combined with either finite difference(FD)or local discontinuous Galerkin(DG)spatial discretization.We analyze the stability of the fully discrete scheme,on a uniform mesh with periodic boundary conditions,using the Fourier method.For the linearized KdV equation,the IMEX schemes are stable under the standard Courant-Friedrichs-Lewy(CFL)conditionτ≤λh.Here,λis the CFL number,τis the time-step size,and h is the spatial mesh size.We study several IMEX schemes and characterize their CFL number as a function ofθ=d/h^(2)with d being the dispersion coefficient,which leads to several interesting observations.We also investigate the asymptotic behaviors of the CFL number for sufficiently refined meshes and derive the necessary conditions for the asymptotic stability of the IMEX-RK methods.Some numerical experiments are provided in the paper to illustrate the performance of IMEX methods under different time-step constraints.

基于Additive Runge-Kutta方法的激波聚焦起爆高精度数值模拟

基于详细氢氧化学动力学模型,建立了描述氢氧爆轰的多组分反应欧拉方程组.针对建立的反应欧拉方程组,数值方法上采用3阶Additive Runge-Kutta方法对时间项进行积分,采用5阶精度的加权本质无振荡(WENO)格式对空间对流项进行离散,自主研发了大规模高精度计算程序.该程序能够处理化学反应源项引起的刚性问题,且能节省计算时间和计算内存.对半球型、半椭球型、圆锥型3种结构形式凹面腔内的激波聚焦起爆过程进行了数值模拟,数值模拟研究得到了不同结构形式凹面腔内的激波聚焦起爆过程.

CONVERGENCE ANALYSIS OF RUNGE-KUTTA METHODS FOR A CLASS OF RETARDED DIFFERENTIAL ALGEBRAIC SYSTEMS

This article deals with a class of numerical methods for retarded differential algebraic systems with time-variable delay. The methods can be viewed as a combination of Runge-Kutta methods and Lagrange interpolation. A new convergence concept, called DA-convergence, is introduced. The DA-convergence result for the methods is derived. At the end, a numerical example is given to verify the computational effectiveness and the theoretical result.

Investigation of feature contribution to shield tunneling-induced settlement using Shapley additive explanations method

Accurate prediction of shield tunneling-induced settlement is a complex problem that requires consideration of many influential parameters.Recent studies reveal that machine learning(ML)algorithms can predict the settlement caused by tunneling.However,well-performing ML models are usually less interpretable.Irrelevant input features decrease the performance and interpretability of an ML model.Nonetheless,feature selection,a critical step in the ML pipeline,is usually ignored in most studies that focused on predicting tunneling-induced settlement.This study applies four techniques,i.e.Pearson correlation method,sequential forward selection(SFS),sequential backward selection(SBS)and Boruta algorithm,to investigate the effect of feature selection on the model’s performance when predicting the tunneling-induced maximum surface settlement(S_(max)).The data set used in this study was compiled from two metro tunnel projects excavated in Hangzhou,China using earth pressure balance(EPB)shields and consists of 14 input features and a single output(i.e.S_(max)).The ML model that is trained on features selected from the Boruta algorithm demonstrates the best performance in both the training and testing phases.The relevant features chosen from the Boruta algorithm further indicate that tunneling-induced settlement is affected by parameters related to tunnel geometry,geological conditions and shield operation.The recently proposed Shapley additive explanations(SHAP)method explores how the input features contribute to the output of a complex ML model.It is observed that the larger settlements are induced during shield tunneling in silty clay.Moreover,the SHAP analysis reveals that the low magnitudes of face pressure at the top of the shield increase the model’s output。

Projected Runge-Kutta methods for constrained Hamiltonian systems

Projected Runge-Kutta （R-K） methods for constrained Hamiltonian systems are proposed. Dynamic equations of the systems, which are index-3 differential-algebraic equations （DAEs） in the Heisenberg form, are established under the framework of Lagrangian multipliers. R-K methods combined with the technique of projections are then used to solve the DAEs. The basic idea of projections is to eliminate the constraint violations at the position, velocity, and acceleration levels, and to preserve the total energy of constrained Hamiltonian systems by correcting variables of the position, velocity, acceleration, and energy. Numerical results confirm the validity and show the high precision of the proposed method in preserving three levels of constraints and total energy compared with results reported in the literature.

The improved artificial bee colony algorithm for mixed additive and multiplicative random error model and the bootstrap method for its precision estimation

To solve the complex weight matrix derivative problem when using the weighted least squares method to estimate the parameters of the mixed additive and multiplicative random error model(MAM error model),we use an improved artificial bee colony algorithm without derivative and the bootstrap method to estimate the parameters and evaluate the accuracy of MAM error model.The improved artificial bee colony algorithm can update individuals in multiple dimensions and improve the cooperation ability between individuals by constructing a new search equation based on the idea of quasi-affine transformation.The experimental results show that based on the weighted least squares criterion,the algorithm can get the results consistent with the weighted least squares method without multiple formula derivation.The parameter estimation and accuracy evaluation method based on the bootstrap method can get better parameter estimation and more reasonable accuracy information than existing methods,which provides a new idea for the theory of parameter estimation and accuracy evaluation of the MAM error model.

Multi-symplectic Runge-Kutta methods for Landau-Ginzburg-Higgs equation

Nonlinear wave equations have been extensively investigated in the last sev- eral decades. The Landau-Ginzburg-Higgs equation, a typical nonlinear wave equation, is studied in this paper based on the multi-symplectic theory in the Hamilton space. The multi-symplectic Runge-Kutta method is reviewed, and a semi-implicit scheme with certain discrete conservation laws is constructed to solve the first-order partial differential equations （PDEs） derived from the Landau-Ginzburg-Higgs equation. The numerical re- sults for the soliton solution of the Landau-Ginzburg-Higgs equation are reported, showing that the multi-symplectic Runge-Kutta method is an efficient algorithm with excellent long-time numerical behaviors.

A class of twostep continuity Runge-Kutta methods for solving singular delay differential equations and its convergence

An idea of relaxing the effect of delay when computing the Runge-Kutta stages in the current step and a class of two-step continuity Runge-Kutta methods （TSCRK） is presented. Their construction, their order conditions and their convergence are studied. The two-step continuity Runge-Kutta methods possess good numerical stability properties and higher stage-order, and keep the explicit process of computing the Runge-Kutta stages. The numerical experiments show that the TSCRK methods are efficient.

Delay-dependent stability analysis of Runge-Kutta methods for neutral delay differential equations

The aim of this paper is to study the asymptotic stability properties of Runge Kutta(R-K) methods for neutral differential equations(NDDEs) when they are applied to the linear test equation of the form: y′(t)=ay(t)+by(t-τ)+cy’(t-τ), t>0, y(t)=g(t), -τ≤t≤0, with a,b,c∈[FK(W+3mm\.3mm][TPP129A,+3mm?3mm,BP], τ>0 and g(t) is a continuous real value function. In this paper we are concerned with the dependence of stability region on a fixed but arbitrary delay τ. In fact, it is one of the N.Guglielmi open problems to investigate the delay dependent stability analysis for NDDEs. The results that the 2,3 stages non natural R-K methods are unstable as Radau IA and Lobatto IIIC are proved. And the s stages Radau IIA methods are unstable, however all Gauss methods are compatible.

A Simple Additive Weighting Method for Time-SeriesMultiindices Decision Making and Its Applications

This paper takes the evaluation of overall economic benefit by an example and proposes a simple additive weighting method for time-series multiindices decision making. The method can automatically determine the weight coefficients among the multiindices and the years respectively and it also can obtain the objective evaluation results and conclusions.

Numerical Stability and Oscillations of Runge-Kutta Methods for Differential Equations with Piecewise Constant Arguments of Advanced Type

For differential equations with piecewise constant arguments of advanced type, numerical stability and oscillations of Runge-Kutta methods are investigated. The necessary and sufficient conditions under which the numerical stability region contains the analytic stability region are given. The conditions of oscillations for the Runge-Kutta methods are obtained also. We prove that the Runge-Kutta methods preserve the oscillations of the analytic solution. Moreover, the relationship between stability and oscillations is discussed. Several numerical examples which confirm the results of our analysis are presented.

Parallel iteration methods of Runge-Kutta methods for delay differential equations

This paper deals with the parallel diagonal implicit Runge-Kutta methods for solving DDEs with a constant delay. It is shown that the suitable choice of the predictor matrix can guarantee the stability of the methods. It is proved that for the suitable selection of the diagonal matrix D, the method based on Radau IIA is δ-convergent, and the estimates for the non-stiff speed and the stiff speed of convergence are given.

Three-stage Stiffly Accurate Runge-Kutta Methods for Stiff Stochastic Differential Equations

In this paper we discuss diagonally implicit and semi-implicit methods based on the three-stage stiffly accurate Runge-Kutta methods for solving Stratonovich stochastic differential equations（SDEs）.Two methods,a three-stage stiffly accurate semi-implicit（SASI3） method and a three-stage stiffly accurate diagonally implicit （SADI3） method,are constructed in this paper.In particular,the truncated random variable is used in the implicit method.The stability properties and numerical results show the effectiveness of these methods in the pathwise approximation of stiff SDEs.

H-stability of the Runge-Kutta methods with general variable stepsize for system of pantograph equations with two delay terms

This paper deals with H-stability of the Runge-Kutta methods with a general variable stepsize for the system of pantograph equations with two delay terms. It is shown that the Runge-Kutta methods with a regular matrix A are H-stable if and only if the modulus of the stability function at infinity is less than 1.

Algebraic Stability of Multistep Runge-Kutta Methods

A series of sufficient and necessary conditions for the algebraic stability of multistepRunge-Kutta methods is obtained, most of which can be regarded as extension of the relevant results available for Runge-Kutta methods, especially, for Radau Ⅰ A, Radau Ⅱ A and Gaussian Runge-Kutta methods.

A Class of Explicit Parallel Multistep Runge-Kutta Methods

In this paper, a rather general class of explicit parallel multistep Runge-Kutta methods is constructed for solving initial value problem of ordinary differential equations. Also, the corresponding convergence and stability are analysed. Several parallel computational formulae are given. The numerical experiments, including accuracy, speedup, and efficiency tests show that the methods are efficient.

Two Implicit Runge-Kutta Methods for Stochastic Differential Equation

In this paper, the Ito-Taylor expansion of stochastic differential equation is briefly introduced. The colored rooted tree theory is applied to derive strong order 1.0 implicit stochastic Runge-Kutta method(SRK). Two fully implicit schemes are presented and their stability qualities are discussed. And the numerical report illustrates the better numerical behavior.

Enforcing Strong Stability of Explicit Runge-Kutta Methods with Superviscosity

A time discretization method is called strongly stable(or monotone),if the norm of its numerical solution is nonincreasing.Although this property is desirable in various of contexts,many explicit Runge-Kutta(RK)methods may fail to preserve it.In this paper,we enforce strong stability by modifying the method with superviscosity,which is a numerical technique commonly used in spectral methods.Our main focus is on strong stability under the inner-product norm for linear problems with possibly non-normal operators.We propose two approaches for stabilization:the modified method and the filtering method.The modified method is achieved by modifying the semi-negative operator with a high order superviscosity term;the filtering method is to post-process the solution by solving a diffusive or dispersive problem with small superviscosity.For linear problems,most explicit RK methods can be stabilized with either approach without accuracy degeneration.Furthermore,we prove a sharp bound(up to an equal sign)on diffusive superviscosity for ensuring strong stability.For nonlinear problems,a filtering method is investigated.Numerical examples with linear non-normal ordinary differential equation systems and for discontinuous Galerkin approximations of conservation laws are performed to validate our analysis and to test the performance.