A Non-Iterative Problem-Dependent Formula for Stiff Dynamic Problems

2021 ◽  
Author(s):  
Shuenn-Yih Chang
Keyword(s):  
Numerical Solutions ◽  
Second Order ◽  
Trapezoidal Rule ◽  
Dynamic Problems ◽  
Step Size ◽  
Order Accuracy ◽  
Explicit Formulation ◽  
One Step ◽  
High Computational Efficiency ◽  
Stability Consideration

Abstract A novel one-step formula is proposed for solving initial value problems based on a concept of eigenmode. It is characterized by problem dependency since it has problem-dependent coefficients, which are functions of the product of the step size and the initial physical properties to define the problem under analysis. It can simultaneously combine A-stability, explicit formulation and second order accuracy. A-stability implies no limitation on step size based on stability consideration. An explicit formulation implies no nonlinear iterations for each step. The second order accuracy with an appropriate step size can have a good accuracy in numerical solutions. Thus, it seems promising for solving stiff dynamic problems. Numerical tests affirm that it can have the same performance as that of the trapezoidal rule for solving linear and nonlinear dynamic problems. It is evident that the most important advantage is of high computational efficiency in contrast to the trapezoidal rule due to no nonlinear iterations of each step.

scholarly journals A Set of New Stable, Explicit, Second Order Schemes for the Non-Stationary Heat Conduction Equation

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2284
Author(s):  
Endre Kovács ◽  
Ádám Nagy ◽  
Mahmoud Saleh
Keyword(s):  
Numerical Solutions ◽  
Finite Difference Methods ◽  
Second Order ◽  
Stiff Systems ◽  
Implicit Methods ◽  
Step Size ◽  
Time Step ◽  
Two Stage ◽  
Time Step Size ◽  
New Methods

This paper introduces a set of new fully explicit numerical algorithms to solve the spatially discretized heat or diffusion equation. After discretizing the space and the time variables according to conventional finite difference methods, these new methods do not approximate the time derivatives by finite differences, but use a combined two-stage constant-neighbour approximation to decouple the ordinary differential equations and solve them analytically. In the final expression for the new values of the variable, the time step size appears not in polynomial or rational, but in exponential form with negative coefficients, which can guarantee stability. The two-stage scheme contains a free parameter p and we analytically prove that the convergence is second order in the time step size for all values of p and the algorithm is unconditionally stable if p is at least 0.5, not only for the linear heat equation, but for the nonlinear Fisher’s equation as well. We compare the performance of the new methods with analytical and numerical solutions. The results suggest that the new algorithms can be significantly faster than the widely used explicit or implicit methods, particularly in the case of extremely large stiff systems.

scholarly journals NUMERICAL SOLUTION FOR SOLVING SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS USING BLOCK METHOD

2012 ◽  
Vol 09 ◽  
pp. 560-565 ◽  
Author(s):  
NUR ZAHIDAH MUKHTAR ◽  
ZANARIAH ABDUL MAJID ◽  
FUDZIAH ISMAIL ◽  
MOHAMED SULEIMAN
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fluid Dynamic ◽  
Electric Circuit ◽  
Second Order ◽  
Variable Step Size ◽  
Step Size ◽  
Step Method ◽  
Mathematical Problems ◽  
One Step

The purpose of this paper is to present a four point direct block one-step method for solving directly the general second order nonstiff initial value problems (IVPs) of ordinary differential equations (ODEs). The mathematical problems in real world can be written in the form of differential equations and arise in the fields of science and engineering such as fluid dynamic, electric circuit, motion of rocket or satellite and other area of application. The proposed method will estimate the approximation solutions at four points simultaneously by using variable step size. Numerical results are given to show the efficiency of the proposed method.

scholarly journals Direct Two-Point Block One-Step Method for Solving General Second-Order Ordinary Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Zanariah Abdul Majid ◽  
Nur Zahidah Mokhtar ◽  
Mohamed Suleiman
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Multistep Method ◽  
Second Order ◽  
Variable Step Size ◽  
Step Size ◽  
Step Method ◽  
One Step ◽  
The Stability ◽  
The One

A direct two-point block one-step method for solving general second-order ordinary differential equations (ODEs) directly is presented in this paper. The one-step block method will solve the second-order ODEs without reducing to first-order equations. The direct solutions of the general second-order ODEs will be calculated at two points simultaneously using variable step size. The method is formulated using the linear multistep method, but the new method possesses the desirable feature of the one-step method. The implementation is based on the predictor and corrector formulas in thePE(CE)mmode. The stability and precision of this method will also be analyzed and deliberated. Numerical results are given to show the efficiency of the proposed method and will be compared with the existing method.

AN ACOUSTIC FINITE-DIFFERENCE TIME-DOMAIN ALGORITHM WITH ISOTROPIC DISPERSION

2005 ◽  
Vol 13 (02) ◽  
pp. 365-384 ◽  
Author(s):  
CHRISTOPHER L. WAGNER ◽  
JOHN B. SCHNEIDER
Keyword(s):  
Finite Difference ◽  
Time Domain ◽  
Finite Difference Time Domain ◽  
Phase Error ◽  
Three Dimensional ◽  
Second Order ◽  
Step Size ◽  
Order Accuracy ◽  
Discretization Step ◽  
Difference Time

The classic Yee Finite-Difference Time-Domain (FDTD) algorithm employs central differences to achieve second-order accuracy, i.e., if the spatial and temporal step sizes are reduced by a factor of n, the phase error associated with propagation through the grid will be reduced by a factor of n2. The Yee algorithm is also second-order isotropic meaning the error as a function of the direction of propagation has a leading term which depends on the square of the discretization step sizes. An FDTD algorithm is presented here that has second-order accuracy but fourth-order isotropy. This algorithm permits a temporal step size 50% larger than that of the three-dimensional Yee algorithm. Pressure-release resonators are used to demonstrate the behavior of the algorithm and to compare it with the Yee algorithm. It is demonstrated how the increased isotropy enables post-processing of the simulation spectra to correct much of the dispersion error. The algorithm can also be optimized at a specified frequency, substantially reducing numerical errors at that design frequency. Also considered are simulations of scattering from penetrable spheres ensonified by a pulsed plane wave. Each simulation yields results at multiple frequencies which are compared to the exact solution. In general excellent agreement is obtained.

Fourth-Order Spatial and Second-Order Temporal Accurate Compact Scheme for Cahn–Hilliard Equation

2019 ◽  
Vol 20 (2) ◽  
pp. 137-143
Author(s):  
Seunggyu Lee
Keyword(s):  
Fourth Order ◽  
Mass Conservation ◽  
Second Order ◽  
High Order Accuracy ◽  
Step Size ◽  
Order Accuracy ◽  
Central Difference ◽  
Hilliard Equation ◽  
Temporal And Spatial ◽  
Cahn Hilliard Equation

AbstractWe propose a fourth-order spatial and second-order temporal accurate and unconditionally stable compact finite-difference scheme for the Cahn–Hilliard equation. The proposed scheme has a higher-order accuracy in space than conventional central difference schemes even though both methods use a three-point stencil. Its compactness may be useful when applying the scheme to numerical implementation. In a temporal discretization, the secant-type algorithm, which is known as the second-order accurate scheme, is applied. Furthermore, the unique solvability regardless of the temporal and spatial step size, unconditionally gradient stability, and discrete mass conservation are proven. It guarantees that large temporal and spatial step sizes could be used with the high-order accuracy and the original properties of the CH equation. Then, numerical results are presented to confirm the efficiency and accuracy of the proposed scheme. The efficiency of the proposed scheme is better than other low order accurate stable schemes.

scholarly journals A Trigonometrically Fitted Block Method for Solving Oscillatory Second-Order Initial Value Problems and Hamiltonian Systems

2017 ◽  
Vol 2017 ◽  
pp. 1-14 ◽  
Author(s):  
F. F. Ngwane ◽  
S. N. Jator
Keyword(s):  
Hamiltonian Systems ◽  
Initial Value Problems ◽  
Grid Point ◽  
Second Order ◽  
Initial Value ◽  
Step Size ◽  
Energy Conserving ◽  
One Step ◽  
The Stability ◽  
Predictor Corrector

In this paper, we present a block hybrid trigonometrically fitted Runge-Kutta-Nyström method (BHTRKNM), whose coefficients are functions of the frequency and the step-size for directly solving general second-order initial value problems (IVPs), including Hamiltonian systems such as the energy conserving equations and systems arising from the semidiscretization of partial differential equations (PDEs). Four discrete hybrid formulas used to formulate the BHTRKNM are provided by a continuous one-step hybrid trigonometrically fitted method with an off-grid point. We implement BHTRKNM in a block-by-block fashion; in this way, the method does not suffer from the disadvantages of requiring starting values and predictors which are inherent in predictor-corrector methods. The stability property of the BHTRKNM is discussed and the performance of the method is demonstrated on some numerical examples to show accuracy and efficiency advantages.

scholarly journals An efficient hybrid numerical scheme for solvinggeneral second order initial value problems (IVPs)

2015 ◽  
Vol 4 (2) ◽  
pp. 411 ◽  
Author(s):  
Oluwadare Adeniran ◽  
Babatunde Ogundare
Keyword(s):  
Initial Value Problems ◽  
Numerical Scheme ◽  
Second Order ◽  
Variable Step Size ◽  
Initial Value ◽  
Step Size ◽  
Interpolation Technique ◽  
One Step ◽  
Grid Points ◽  
Order Four

<p>The paper presents a one step hybrid numerical scheme with two off grid points for solving directly the general second order initial value problems of ordinary differential equations. The scheme is developed using collocation and interpolation technique. The proposed scheme is consistent, zero stable and of order four. This scheme can estimate the approximate solution at both step and off step points simultaneously by using variable step size. Numerical results are given to show the efficiency of the proposed scheme over the existing schemes.</p>

scholarly journals Meta-Descent for Online, Continual Prediction

2019 ◽  
Vol 33 ◽  
pp. 3943-3950
Author(s):  
Andrew Jacobsen ◽  
Matthew Schlegel ◽  
Cameron Linke ◽  
Thomas Degris ◽  
Adam White ◽  
...  
Keyword(s):  
Gradient Descent ◽  
Large Scale ◽  
Time Series Prediction ◽  
Real Data ◽  
Second Order ◽  
Stochastic Gradient Descent ◽  
Step Size ◽  
Vector Approximation ◽  
Prediction Problems ◽  
Stationary Problems

This paper investigates different vector step-size adaptation approaches for non-stationary online, continual prediction problems. Vanilla stochastic gradient descent can be considerably improved by scaling the update with a vector of appropriately chosen step-sizes. Many methods, including AdaGrad, RMSProp, and AMSGrad, keep statistics about the learning process to approximate a second order update—a vector approximation of the inverse Hessian. Another family of approaches use meta-gradient descent to adapt the stepsize parameters to minimize prediction error. These metadescent strategies are promising for non-stationary problems, but have not been as extensively explored as quasi-second order methods. We first derive a general, incremental metadescent algorithm, called AdaGain, designed to be applicable to a much broader range of algorithms, including those with semi-gradient updates or even those with accelerations, such as RMSProp. We provide an empirical comparison of methods from both families. We conclude that methods from both families can perform well, but in non-stationary prediction problems the meta-descent methods exhibit advantages. Our method is particularly robust across several prediction problems, and is competitive with the state-of-the-art method on a large-scale, time-series prediction problem on real data from a mobile robot.

scholarly journals Verification of Convergence Rates of Numerical Solutions for Parabolic Equations

2019 ◽  
Vol 2019 ◽  
pp. 1-10
Author(s):  
Darae Jeong ◽  
Yibao Li ◽  
Chaeyoung Lee ◽  
Junxiang Yang ◽  
Yongho Choi ◽  
...  
Keyword(s):  
Parabolic Equations ◽  
Convergence Rates ◽  
Numerical Solutions ◽  
Second Order ◽  
Order Convergence ◽  
Numerical Convergence ◽  
Verification Method ◽  
First Order ◽  
Convergence Test ◽  
Second Order Convergence

In this paper, we propose a verification method for the convergence rates of the numerical solutions for parabolic equations. Specifically, we consider the numerical convergence rates of the heat equation, the Allen–Cahn equation, and the Cahn–Hilliard equation. Convergence test results show that if we refine the spatial and temporal steps at the same time, then we have the second-order convergence rate for the second-order scheme. However, in the case of the first-order in time and the second-order in space scheme, we may have the first-order or the second-order convergence rates depending on starting spatial and temporal step sizes. Therefore, for a rigorous numerical convergence test, we need to perform the spatial and the temporal convergence tests separately.

An efficient variable step-size rational Falkner-type method for solving the special second-order IVP

2016 ◽  
Vol 291 ◽  
pp. 39-51 ◽  
Author(s):  
Higinio Ramos ◽  
Gurjinder Singh ◽  
V. Kanwar ◽  
Saurabh Bhatia
Keyword(s):  
Second Order ◽  
Variable Step Size ◽  
Step Size ◽  
Type Method ◽  
Variable Step
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