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.

A Strategy to Accelerate the Numerical Integration in the Dynamic Simulation of Multibody Systems

1997 ◽  
Author(s):  
Jesús Cardenal ◽  
Javier Cuadrado ◽  
Eduardo Bayo
Keyword(s):  
Multibody Systems ◽  
Equations Of Motion ◽  
Stiff Systems ◽  
Step Size ◽  
Step Method ◽  
Integral Of Motion ◽  
Time Step ◽  
Time Step Size ◽  
Variable Time ◽  
Numerical Integrator

Abstract This paper presents a multi-index variable time step method for the integration of the equations of motion of constrained multibody systems in descriptor form. The basis of the method is the augmented Lagrangian formulation with projections in index-3 and index-1. The method takes advantage of the better performance of the index-3 formulation for large time steps and of the stability of the index-1 for low time steps, and automatically switches from one method to the other depending on the required accuracy and values of the time step. The variable time stepping is accomplished through the use of an integral of motion, which in the case of conservative systems becomes the total energy. The error introduced by the numerical integrator in the integral of motion during consecutive time steps provides a good measure of the local integration error, and permits a simple and reliable strategy for varying the time step. Overall, the method is efficient and powerful; it is suitable for stiff and non-stiff systems, robust for all time step sizes, and it works for singular configurations, redundant constraints and topology changes. Also, the constraints in positions, velocities and accelerations are satisfied during the simulation process. The method is robust in the sense that becomes more accurate as the time step size decreases.

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

2010 ◽  
Vol 138 (8) ◽  
pp. 3333-3341 ◽  
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.

Horizontally Explicit and Vertically Implicit (HEVI) Time Discretization Scheme for a Discontinuous Galerkin Nonhydrostatic Model

2015 ◽  
Vol 143 (3) ◽  
pp. 972-990 ◽  
Author(s):  
Lei Bao ◽  
Robert Klöfkorn ◽  
Ramachandran D. Nair
Keyword(s):  
Discontinuous Galerkin ◽  
Strong Stability ◽  
Euler System ◽  
Second Order ◽  
Spatial Discretization ◽  
Step Size ◽  
Time Step ◽  
Exact Integration ◽  
Time Step Size ◽  
Propagating Waves

Abstract A two-dimensional nonhydrostatic (NH) atmospheric model based on the compressible Euler system has been developed in the (x, z) Cartesian domain. The spatial discretization is based on a nodal discontinuous Galerkin (DG) method with exact integration. The orography is handled by the terrain-following height-based coordinate system. The time integration uses the horizontally explicit and vertically implicit (HEVI) time-splitting scheme, which is introduced to address the stringent restriction on the explicit time step size due to a high aspect ratio between the horizontal (x) and vertical (z) spatial discretization. The HEVI scheme is generally based on the Strang-type operator-split approach, where the horizontally propagating waves in the Euler system are solved explicitly while the vertically propagating waves are treated implicitly. As a consequence, the HEVI scheme relaxes the maximum allowed time step to be mainly determined by the horizontal grid spacing. The accuracy of the HEVI scheme is rigorously compared against that of the explicit strong stability-preserving (SSP) Runge–Kutta (RK) scheme using several NH benchmark test cases. The HEVI scheme shows a second-order temporal convergence, as expected. The results of the HEVI scheme are qualitatively comparable to those of the SSP-RK3 scheme. Moreover, the HEVI DG formulation can also be seamlessly extended to account for the second-order diffusion as in the case of the standard SSP-RK DG formulation. In the presence of orography, the HEVI scheme produces high quality results, which are visually identical to those produced by the SSP-RK3 scheme.

Seismic finite‐difference modeling with spatially varying time steps

Geophysics ◽  
2000 ◽  
Vol 65 (4) ◽  
pp. 1290-1293 ◽  
Author(s):  
Ekkehart Tessmer
Keyword(s):  
Finite Difference ◽  
Seismic Velocity ◽  
Finite Difference Methods ◽  
Low Velocity ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Large Velocity ◽  
Numerical Grid ◽  
Spatially Varying

Numerical seismic modeling by finite‐difference methods usually works with a global time‐step size. Because of stability considerations, the time‐step size is determined essentially by the highest seismic velocity, i.e., the higher the highest velocity, the smaller the time step needs to be. Therefore, if large velocity contrasts exist within the numerical grid, domains of low velocity are oversampled temporally. Using different time‐step sizes in different parts of the numerical grid can reduce computational costs considerably.

scholarly journals A Lie group variational integration approach to the full discretization of a constrained geometrically exact Cosserat beam model

2021 ◽  
Author(s):  
Stefan Hante ◽  
Denise Tumiotto ◽  
Martin Arnold
Keyword(s):  
Lie Group ◽  
Group Structure ◽  
Equations Of Motion ◽  
Second Order ◽  
The Body ◽  
Benchmark Problems ◽  
Beam Model ◽  
Step Size ◽  
Time Step ◽  
Time Step Size

AbstractIn this paper, we will consider a geometrically exact Cosserat beam model taking into account the industrial challenges. The beam is represented by a framed curve, which we parametrize in the configuration space $\mathbb{S}^{3}\ltimes \mathbb{R}^{3}$ S 3 ⋉ R 3 with semi-direct product Lie group structure, where $\mathbb{S}^{3}$ S 3 is the set of unit quaternions. Velocities and angular velocities with respect to the body-fixed frame are given as the velocity vector of the configuration. We introduce internal constraints, where the rigid cross sections have to remain perpendicular to the center line to reduce the full Cosserat beam model to a Kirchhoff beam model. We derive the equations of motion by Hamilton’s principle with an augmented Lagrangian. In order to fully discretize the beam model in space and time, we only consider piecewise interpolated configurations in the variational principle. This leads, after approximating the action integral with second order, to the discrete equations of motion. Here, it is notable that we allow the Lagrange multipliers to be discontinuous in time in order to respect the derivatives of the constraint equations, also known as hidden constraints. In the last part, we will test our numerical scheme on two benchmark problems that show that there is no shear locking observable in the discretized beam model and that the errors are observed to decrease with second order with the spatial step size and the time step size.

An Improved Lattice Boltzmann Method for Steady Fluid Flows

2004 ◽  
Author(s):  
Aditya C. Velivelli ◽  
Kenneth M. Bryden
Keyword(s):  
Lattice Boltzmann Method ◽  
Lattice Boltzmann ◽  
Finite Difference Methods ◽  
Nonlinear Partial Differential Equation ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Flow Problems ◽  
Computational Performance ◽  
Boltzmann Method

The use of the lattice Boltzmann method in computational fluid dynamics has been steadily increasing. The highly local nature of lattice Boltzmann computations have allowed for easy cache optimization and parallelization. This bestows the lattice Boltzmann method with considerable superiority in computational performance over traditional finite difference methods for solving unsteady flow problems. When solving steady flow problems, the explicit nature of the lattice Boltzmann discretization limits the time step size. The time step size is limited by the Courant-Friedrichs-Lewy (CFL) condition and local gradients in the solution, the latter limitation being more extreme. This paper describes a novel explicit discretization for the lattice Boltzmann method that can perform simulations with larger time step sizes. The new algorithm is applid to the steady Burger’s equation, uux = μ(uxx + uyy), which is a nonlinear partial differential equation containing both convection and diffusion terms. A comparison between the original lattice Boltzmann method and the new algorithm is performed with regard to time for computation and accuracy.

Traffic Flow Simulation through Parallel Processing

1996 ◽  
Vol 1566 (1) ◽  
pp. 31-38 ◽  
Author(s):  
Anthony Theodore Chronopoulos ◽  
Gang Wang
Keyword(s):  
Numerical Methods ◽  
Traffic Flow ◽  
Traffic Simulation ◽  
Parallel Computer ◽  
Implicit Methods ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
The Past ◽  
Simulation Codes

Numerical methods for solving traffic flow continuum models have been studied and efficiently implemented in traffic simulation codes in the past. Explicit and implicit methods have been used in traffic simulation codes in the past. Implicit methods allow a much larger time step size than explicit methods to achieve the same accuracy. However, at each time step a nonlinear system must be solved. The Newton method, coupled with a linear iterative method (Orthomin), is used. The efficient implementation of explicit and implicit numerical methods for solving the high-order flow conservation traffic model on parallel computers was studied. Simulation tests were run with traffic data from an 18-mile freeway section in Minnesota on the nCUBE2 parallel computer. These tests gave the same accuracy as past tests, which were performed on one-processor computers, and the overall execution time was significantly reduced.

scholarly journals A Nonoscillatory Second-Order Time-Stepping Procedure for Reaction-Diffusion Equations

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Philku Lee ◽  
George V. Popescu ◽  
Seongjai Kim
Keyword(s):  
Numerical Solution ◽  
Reaction Diffusion ◽  
Second Order ◽  
Reaction Diffusion Equations ◽  
Step Size ◽  
Time Step ◽  
Potential Oscillations ◽  
Time Step Size ◽  
Time Stepping ◽  
Spurious Oscillations

After a theory of morphogenesis in chemical cells was introduced in the 1950s, much attention had been devoted to the numerical solution of reaction-diffusion (RD) partial differential equations (PDEs). The Crank–Nicolson (CN) method has been a common second-order time-stepping procedure. However, the CN method may introduce spurious oscillations for nonsmooth data unless the time step size is sufficiently small. This article studies a nonoscillatory second-order time-stepping procedure for RD equations, called a variable-θmethod, as a perturbation of the CN method. In each time level, the new method detects points of potential oscillations to implicitly resolve the solution there. The proposed time-stepping procedure is nonoscillatory and of a second-order temporal accuracy. Various examples are given to show effectiveness of the method. The article also performs a sensitivity analysis for the numerical solution of biological pattern forming models to conclude that the numerical solution is much more sensitive to the spatial mesh resolution than the temporal one. As the spatial resolution becomes higher for an improved accuracy, the CN method may produce spurious oscillations, while the proposed method results in stable solutions.

Efficient Time-Stepping/Spectral Methods for the Navier-Stokes-Nernst-Planck-Poisson Equations

2017 ◽  
Vol 21 (5) ◽  
pp. 1408-1428 ◽  
Author(s):  
Xiaoling Liu ◽  
Chuanju Xu
Keyword(s):  
Stability Condition ◽  
Time Discretization ◽  
Equation System ◽  
Second Order ◽  
Navier Stokes ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
First Order ◽  
Time Stepping

AbstractThis paper is concerned with numerical methods for the Navier-Stokes-Nernst-Planck-Poisson equation system. The main goal is to construct and analyze some stable time stepping schemes for the time discretization and use a spectral method for the spatial discretization. The main contribution of the paper includes: 1) an useful stability inequality for the weak solution is derived; 2) a first order time stepping scheme is constructed, and the non-negativity of the concentration components of the discrete solution is proved. This is an important property since the exact solution shares the same property. Moreover, the stability of the scheme is established, together with a stability condition on the time step size; 3) a modified first order scheme is proposed in order to decouple the calculation of the velocity and pressure in the fluid field. This new scheme equally preserves the non-negativity of the discrete concentration solution, and is stable under a similar stability condition; 4) a stabilization technique is introduced to make the above mentioned schemes stable without restriction condition on the time step size; 5) finally we construct a second order finite difference scheme in time and spectral discretization in space. The numerical tests carried out in the paper show that all the proposed schemes possess some desirable properties, such as conditionally/unconditionally stability, first/second order convergence, non-negativity of the discrete concentrations, and so on.

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