On the Stability of a Variable Step Exponential Splitting Method for Solving Multidimensional Quenching-Combustion Equations

2016 ◽  
pp. 155-167 ◽  
Author(s):  
Joshua L. Padgett ◽  
Qin Sheng
Keyword(s):  
Splitting Method ◽  
Variable Step ◽  
Exponential Splitting ◽  
The Stability ◽  
Combustion Equations

Solving Vlasov-Poisson-Fokker-Planck Equations using NRxx method

2017 ◽  
Vol 21 (3) ◽  
pp. 782-807 ◽  
Author(s):  
Yanli Wang ◽  
Shudao Zhang
Keyword(s):  
Splitting Method ◽  
Linear Convergence ◽  
Mass Conservation ◽  
System A ◽  
Spectral Convergence ◽  
The Stability ◽  
The Moment ◽  
Stability And Accuracy ◽  
Fokker Planck Equations ◽  
Fokker Planck

AbstractWe present a numerical method to solve the Vlasov-Poisson-Fokker-Planck (VPFP) system using the NRxx method proposed in [4, 7, 9]. A globally hyperbolic moment system similar to that in [23] is derived. In this system, the Fokker-Planck (FP) operator term is reduced into the linear combination of the moment coefficients, which can be solved analytically under proper truncation. The non-splitting method, which can keep mass conservation and the balance law of the total momentum, is used to solve the whole system. A numerical problem for the VPFP system with an analytic solution is presented to indicate the spectral convergence with the moment number and the linear convergence with the grid size. Two more numerical experiments are tested to demonstrate the stability and accuracy of the NRxx method when applied to the VPFP system.

Variable-step-size second-order-derivative multistep method for solving first-order ordinary differential equations in system simulation

2020 ◽  
Vol 11 (01) ◽  
pp. 2050001
Author(s):  
Lei Zhang ◽  
Chaofeng Zhang ◽  
Mengya Liu
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Multistep Method ◽  
Second Order ◽  
Order Derivative ◽  
Variable Step Size ◽  
Variable Order ◽  
Step Size ◽  
Variable Step ◽  
The Stability

According to the relationship between truncation error and step size of two implicit second-order-derivative multistep formulas based on Hermite interpolation polynomial, a variable-order and variable-step-size numerical method for solving differential equations is designed. The stability properties of the formulas are discussed and the stability regions are analyzed. The deduced methods are applied to a simulation problem. The results show that the numerical method can satisfy calculation accuracy, reduce the number of calculation steps and accelerate calculation speed.

scholarly journals Hybrid Fixed-Point Fixed-Stress Splitting Method for Linear Poroelasticity

Geosciences ◽  
2019 ◽  
Vol 9 (1) ◽  
pp. 29 ◽  
Author(s):  
Paul M. Delgado ◽  
V. M. Krushnarao Kotteda ◽  
Vinod Kumar
Keyword(s):  
Fixed Point ◽  
Asymptotic Solution ◽  
Relative Error ◽  
Operator Splitting ◽  
Splitting Method ◽  
Approximate Solutions ◽  
Time Step ◽  
Operator Splitting Method ◽  
The Stability ◽  
Stability And Consistency

Efficient and accurate poroelasticity models are critical in modeling geophysical problems such as oil exploration, gas-hydrate detection, and hydrogeology. We propose an efficient operator splitting method for Biot’s model of linear poroelasticity based on fixed-point iteration and constrained stress. In this method, we eliminate the constraint on time step via combining our fixed-point approach with a physics-based restraint between iterations. Three different cases are considered to demonstrate the stability and consistency of the method for constant and variable parameters. The results are validated against the results from the fully coupled approach. In case I, a single iteration is used for continuous coefficients. The relative error decreases with an increase in time. In case II, material coefficients are assumed to be linear. In the single iteration approach, the relative error grows significantly to 40% before rapidly decaying to zero. This is an artifact of the approximate solutions approaching the asymptotic solution. The error in the multiple iterations oscillates within 10 − 6 before decaying to the asymptotic solution. Nine iterations per time step are enough to achieve the relative error close to 10 − 7 . In the last case, the hybrid method with multiple iterations requires approximately 16 iterations to make the relative error 5 × 10 − 6 .

scholarly journals New Splitting Methods for Convection-Dominated Diffusion Problems and Navier-Stokes Equations

2014 ◽  
Vol 16 (5) ◽  
pp. 1239-1262 ◽  
Author(s):  
Feng Shi ◽  
Guoping Liang ◽  
Yubo Zhao ◽  
Jun Zou
Keyword(s):  
Stokes Equations ◽  
Stokes Problem ◽  
Splitting Method ◽  
Iterative Solvers ◽  
Diffusion System ◽  
Navier Stokes ◽  
Time Step ◽  
Navier Stokes Equations ◽  
The Stability ◽  
Diffusion Problems

AbstractWe present a new splitting method for time-dependent convention-dominated diffusion problems. The original convention diffusion system is split into two sub-systems: a pure convection system and a diffusion system. At each time step, a convection problem and a diffusion problem are solved successively. A few important features of the scheme lie in the facts that the convection subproblem is solved explicitly and multistep techniques can be used to essentially enlarge the stability region so that the resulting scheme behaves like an unconditionally stable scheme; while the diffusion subproblem is always self-adjoint and coercive so that they can be solved efficiently using many existing optimal preconditioned iterative solvers. The scheme can be extended for solving the Navier-Stokes equations, where the nonlinearity is resolved by a linear explicit multistep scheme at the convection step, while only a generalized Stokes problem is needed to solve at the diffusion step and the major stiffness matrix stays invariant in the time marching process. Numerical simulations are presented to demonstrate the stability, convergence and performance of the single-step and multistep variants of the new scheme.

The stability of variable step-size LMS algorithms

1999 ◽  
Vol 47 (12) ◽  
pp. 3277-3288 ◽  
Author(s):  
S.B. Gelfand ◽  
Yongbin Wei ◽  
J.V. Krogmeier
Keyword(s):  
Variable Step Size ◽  
Step Size ◽  
Variable Step ◽  
The Stability

scholarly journals Variable Step Size Adams Methods for BSDEs

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Qiang Han
Keyword(s):  
High Order ◽  
Variable Step Size ◽  
Necessary And Sufficient Condition ◽  
Step Size ◽  
Truncation Errors ◽  
Adams Methods ◽  
Variable Step ◽  
High Order Convergence ◽  
The Stability ◽  
Necessary And Sufficient

For backward stochastic differential equations (BSDEs), we construct variable step size Adams methods by means of Itô–Taylor expansion, and these schemes are nonlinear multistep schemes. It is deduced that the conditions of local truncation errors with respect to Y and Z reach high order. The coefficients in the numerical methods are inferred and bounded under appropriate conditions. A necessary and sufficient condition is given to judge the stability of our numerical schemes. Moreover, the high-order convergence of the schemes is rigorously proved. The numerical illustrations are provided.

A Class of Methods with Optimal Stability Properties for the Numerical Solution of IVPs: Construction and Implementation

2017 ◽  
Vol 14 (01) ◽  
pp. 1750007
Author(s):  
Masoumeh Hosseini Nasab ◽  
Gholamreza Hojjati ◽  
Ali Abdi
Keyword(s):  
Numerical Solution ◽  
Numerical Experiments ◽  
Point Of View ◽  
Variable Step Size ◽  
Second Derivative ◽  
Step Size ◽  
Stability Properties ◽  
Variable Step ◽  
The Stability ◽  
Size Mode

Considering the methods with future points technique from second derivative general linear methods (SGLMs) point of view, makes it possible to improve their stability properties. In this paper, we extend the stability regions of a modified version of E2BD formulas to optimal one and show its effectiveness by numerical verifications. Also, implementation issues, with numerical experiments, of these methods are investigated in a variable step-size mode.

Stability Analysis of Milling Processes With Periodic Spindle Speed Variation Via the Variable-Step Numerical Integration Method

2016 ◽  
Vol 138 (11) ◽  
Author(s):  
Jinbo Niu ◽  
Ye Ding ◽  
LiMin Zhu ◽  
Han Ding
Keyword(s):  
Stability Analysis ◽  
Numerical Integration ◽  
Integration Method ◽  
Spindle Speed ◽  
Numerical Integration Method ◽  
Speed Variation ◽  
Spindle Speed Variation ◽  
Variable Step ◽  
Modulation Schemes ◽  
The Stability

This paper proposes a general method for the stability analysis and parameter optimization of milling processes with periodic spindle speed variation (SSV). With the aid of Fourier series, the time-variant spindle speeds of different periodic modulation schemes are unified into one framework. Then the time-varying delay is derived implicitly and calculated efficiently using an accurate ordinary differential equation (ODE) based algorithm. After incorporating the unified spindle speed and time delay into the dynamic model, a Floquet theory based variable-step numerical integration method (VNIM) is presented for the stability analysis of variable spindle speed milling processes. By comparison with other methods, such as the semi-discretization method and the constant-step numerical integration method, the proposed method has the advantages of high computational accuracy and efficiency. Finally, different spindle speed modulation schemes are compared and the modulation parameters are optimized with the aid of three-dimensional stability charts obtained using the proposed VNIM.

scholarly journals Iterative Algorithms for Symmetric Positive Semidefinite Solutions of the Lyapunov Matrix Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-10 ◽  
Author(s):  
Min Sun ◽  
Jing Liu
Keyword(s):  
Splitting Method ◽  
Iterative Algorithms ◽  
Positive Semidefinite ◽  
Matrix Equations ◽  
Proximal Point ◽  
Alternating Direction ◽  
Lyapunov Matrix ◽  
Lyapunov Matrix Equations ◽  
The Stability ◽  
Constrained Solutions

It is well-known that the stability of a first-order autonomous system can be determined by testing the symmetric positive definite solutions of associated Lyapunov matrix equations. However, the research on the constrained solutions of the Lyapunov matrix equations is quite few. In this paper, we present three iterative algorithms for symmetric positive semidefinite solutions of the Lyapunov matrix equations. The first and second iterative algorithms are based on the relaxed proximal point algorithm (RPPA) and the Peaceman–Rachford splitting method (PRSM), respectively, and their global convergence can be ensured by corresponding results in the literature. The third iterative algorithm is based on the famous alternating direction method of multipliers (ADMM), and its convergence is subsequently discussed in detail. Finally, numerical simulation results illustrate the effectiveness of the proposed iterative algorithms.

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