Explicit methods for integrating stiff Cauchy problems

An explicit method for solving stiff Cauchy problems is proposed. The method relies on explicit schemes and a step size selection algorithm based on the curvature of an integral curve. Closed-form formulas are derived for finding the curvature. For Runge-Kutta schemes with up to four stages, the corresponding sets of coefficients are given. The method is validated on a test problem with a given exact solution. It is shown that the method is as accurate and robust as implicit methods, but is substantially superior to them in efficiency. A numerical example involving chemical kinetics computations with 9 components and 50 reactions is given.

Download Full-text

On Numerical Methods for Second-Order Nonlinear Ordinary Differential Equations (ODEs): A Reduction To A System Of First-Order ODEs

Universiti Malaysia Terengganu Journal of Undergraduate Research ◽

10.46754/umtjur.v1i4.86 ◽

2019 ◽

Vol 1 (4) ◽

pp. 1-8

Author(s):

Loy Kak Choon ◽

Puteri Nurul Fatihah binti Mohamad Azli

Keyword(s):

Test Problem ◽

Computational Time ◽

Test Cases ◽

Nonlinear Ordinary Differential Equations ◽

Step Size ◽

Nonlinear Odes ◽

First Order ◽

Cpu Time ◽

Explicit Schemes ◽

Manufactured Solution

2nd-order ODEs can be found in many applications, e.g., motion of pendulum, vibrating springs, etc. We first convert the 2nd-order nonlinear ODEs to a system of 1st-order ODEs which is easier to deal with. Then, Adams-Bashforth (AB) methods are used to solve the resulting system of 1st-order ODE. AB methods are chosen since they are the explicit schemes and more efficient in terms of shorter computational time. However, the step size is more restrictive since these methods are conditionally stable. We find two test cases (one test problem and one manufactured solution) to be used to validate the AB methods. The exact solution for both test cases are available for the error and convergence analysis later on. The implementation of 1st-, 2nd- and 3rd-order AB methods are done using Octave. The error was computed to retrieve the order of convergence numerically and the CPU time was recorded to analyze their efficiency.

Download Full-text

An explicit method to calculate implicit spatial finite differences

Geophysics ◽

10.1190/geo2021-0001.1 ◽

2021 ◽

pp. 1-71

Author(s):

Hongwei Liu ◽

Yi Luo

Keyword(s):

Finite Difference ◽

High Performance ◽

Computational Cost ◽

New Method ◽

Seismic Exploration ◽

Explicit Method ◽

Lu Decomposition ◽

Implicit Methods ◽

Band Matrices ◽

Explicit Finite Difference

The finite-difference solution of the second-order acoustic wave equation is a fundamental algorithm in seismic exploration for seismic forward modeling, imaging, and inversion. Unlike the standard explicit finite difference (EFD) methods that usually suffer from the so-called "saturation effect", the implicit FD methods can obtain much higher accuracy with relatively short operator length. Unfortunately, these implicit methods are not widely used because band matrices need to be solved implicitly, which is not suitable for most high-performance computer architectures. We introduce an explicit method to overcome this limitation by applying explicit causal and anti-causal integrations. We can prove that the explicit solution is equivalent to the traditional implicit LU decomposition method in analytical and numerical ways. In addition, we also compare the accuracy of the new methods with the traditional EFD methods up to 32nd order, and numerical results indicate that the new method is more accurate. In terms of the computational cost, the newly proposed method is standard 8th order EFD plus two causal and anti-causal integrations, which can be applied recursively, and no extra memory is needed. In summary, compared to the standard EFD methods, the new method has a spectral-like accuracy; compared to the traditional LU-decomposition implicit methods, the new method is explicit. It is more suitable for high-performance computing without losing any accuracy.

Download Full-text

On the Solution of the Diffusion Equations by Numerical Methods

Journal of Heat Transfer ◽

10.1115/1.3691590 ◽

1966 ◽

Vol 88 (4) ◽

pp. 421-427 ◽

Cited By ~ 96

Author(s):

H. Z. Barakat ◽

J. A. Clark

Keyword(s):

Exact Solution ◽

Finite Difference ◽

Present Method ◽

Finite Difference Methods ◽

Finite Difference Approximation ◽

Difference Approximation ◽

Approximation Procedure ◽

Implicit Methods ◽

Difference Methods ◽

Explicit Finite Difference

An explicit-finite difference approximation procedure which is unconditionally stable for the solution of the general multidimensional, nonhomogeneous diffusion equation is presented. This method possesses the advantages of the implicit methods, i.e., no severe limitation on the size of the time increment. Also it has the simplicity of the explicit methods and employs the same “marching” type technique of solution. Results obtained by this method for several different problems are compared with the exact solution and with those obtained by other finite-difference methods. For the examples solved the numerical results obtained by the present method are in closer agreement with the exact solution than are those obtained by the other methods.

Download Full-text

Accuracy Analysis of a Spectral Element Atmospheric Model Using a Fully Implicit Solution Framework

Monthly Weather Review ◽

10.1175/2010mwr3288.1 ◽

2010 ◽

Vol 138 (8) ◽

pp. 3333-3341 ◽

Cited By ~ 19

Author(s):

Katherine J. Evans ◽

Mark A. Taylor ◽

John B. Drake

Keyword(s):

Shallow Water Equation ◽

Time Integration ◽

Atmospheric Model ◽

Test Cases ◽

Implicit Methods ◽

Step Size ◽

Time Step ◽

Time Step Size ◽

Fully Implicit ◽

Stability Limits

Abstract A fully implicit (FI) time integration method has been implemented into a spectral finite-element shallow-water equation model on a sphere, and it is compared to existing fully explicit leapfrog and semi-implicit methods for a suite of test cases. This experiment is designed to determine the time step sizes that minimize simulation time while maintaining sufficient accuracy for these problems. For test cases without an analytical solution from which to compare, it is demonstrated that time step sizes 30–60 times larger than the gravity wave stability limits and 6–20 times larger than the advective-scale stability limits are possible using the FI method without a loss in accuracy, depending on the problem being solved. For a steady-state test case, the FI method produces error within machine accuracy limits as with existing methods, but using an arbitrarily large time step size.

Download Full-text

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

Mathematics ◽

10.3390/math9182284 ◽

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.

Download Full-text

Implicit Methods in Combustion and Chemical Kinetics Modeling

Multiple Time Scales ◽

10.1016/b978-0-12-123420-1.50010-0 ◽

1985 ◽

pp. 113-144 ◽

Cited By ~ 19

Author(s):

ROBERT J. KEE ◽

LINDA R. PETZOLD ◽

MITCHELL D. SMOOKE ◽

JOSEPH F. GRCAR

Keyword(s):

Chemical Kinetics ◽

Implicit Methods ◽

Kinetics Modeling

Download Full-text

A Quantitative Assessment of the Potential of Implicit Integration Methods for Molecular Dynamics Simulation

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4001392 ◽

2010 ◽

Vol 5 (3) ◽

Author(s):

Nick Schafer ◽

Dan Negrut

Keyword(s):

Molecular Dynamics ◽

Md Simulation ◽

Time Integration ◽

Dynamics Simulation ◽

Integration Step ◽

Implicit Integration ◽

Implicit Methods ◽

Current Standard ◽

Step Size ◽

Computational Overhead

Implicit integration, unencumbered by numerical stability constraints, is attractive in molecular dynamics (MD) simulation due to its presumed ability to advance the simulation at large step sizes. It is not clear what step size values can be expected and if the larger step sizes will compensate for the computational overhead associated with an implicit integration method. The goal of this paper is to answer these questions and thereby assess quantitatively the potential of implicit integration in MD. Two implicit methods (midpoint and Hilber–Hughes–Taylor) are compared with the current standard for MD time integration (explicit velocity Verlet). The implicit algorithms were implemented in a research grade MD code, which used a first-principles interaction potential for biological molecules. The nonlinear systems of equations arising from the use of implicit methods were solved in a quasi-Newton framework. Aspects related to a Newton–Krylov type method are also briefly discussed. Although the energy conservation provided by the implicit methods was good, the integration step size lengths were limited by loss of convergence in the Newton iteration. Moreover, a spectral analysis of the dynamic response indicated that high frequencies present in the velocity and acceleration signals prevent a substantial increase in integration step size lengths. The overhead associated with implicit integration prevents this class of methods from having a decisive impact in MD simulation, a conclusion supported by a series of quantitative analyses summarized in the paper.

Download Full-text

A simple step size selection algorithm for ODE codes

Journal of Computational and Applied Mathematics ◽

10.1016/0377-0427(94)00007-n ◽

1995 ◽

Vol 58 (3) ◽

pp. 345-354 ◽

Cited By ~ 12

Author(s):

L.F. Shampine ◽

A. Witt

Keyword(s):

Selection Algorithm ◽

Step Size ◽

Size Selection ◽

Simple Step

Download Full-text

Explicit and Implicit Methods for Design Sensitivity Analysis of Nonlinear Structures Under Dynamic Loads

Applied Mechanics Reviews ◽

10.1115/1.3101823 ◽

1997 ◽

Vol 50 (11S) ◽

pp. S11-S19 ◽

Cited By ~ 7

Author(s):

Jasbir S. Arora ◽

Abhijit Dutta

Keyword(s):

Sensitivity Analysis ◽

Stiffness Matrix ◽

Design Sensitivity ◽

Implicit Method ◽

Design Sensitivity Analysis ◽

Dynamic Loads ◽

Nonlinear Structures ◽

Explicit Method ◽

Implicit Methods ◽

Analysis Phase

Explicit and implicit methods for design sensitivity analysis of nonlinear structures subjected to transient dynamic loads are described. The direct differentiation method is used to calculate the sensitivities. Elastic and elasto-plastic problems undergoing large deformation are treated in the formulation. The explicit method is quite straightforward to implement for analysis as well as design sensitivity analysis: knowing all the quantities at time t, one updates the quantities for the next time point without solution of any linear system of equations, or any iterations. For the implicit method, the effective stiffness matrix for the sensitivity equation is different from that for the analysis phase due to inclusion of large deformation in the formulation. A numerical procedure to integrate the equation is proposed which uses the symmetric effective stiffness matrix from the analysis phase. Although the sensitivity analysis problem is linear, the procedure requires iterations for sensitivity calculations at each time step, as for the analysis phase. The implicit method is implemented using the pre-conditioned conjugate gradient iterative solution scheme. Numerical results for sensitivities indicate the implicit method to be more accurate than the explicit method for transient elastoplastic problems. However, the explicit method is substantially more efficient than the implicit method.

Download Full-text