scholarly journals Error Estimates of a Continuous Galerkin Time Stepping Method for Subdiffusion Problem

2021 ◽  
Vol 88 (3) ◽  
Author(s):  
Yuyuan Yan ◽  
Bernard A. Egwu ◽  
Zongqi Liang ◽  
Yubin Yan
Keyword(s):  
Piecewise Linear ◽  
Time Discretization ◽  
Convergence Order ◽  
Step Size ◽  
Time Step ◽  
Transform Method ◽  
Nonsmooth Data ◽  
Time Step Size ◽  
Time Stepping ◽  
Continuous Galerkin

AbstractA continuous Galerkin time stepping method is introduced and analyzed for subdiffusion problem in an abstract setting. The approximate solution will be sought as a continuous piecewise linear function in time t and the test space is based on the discontinuous piecewise constant functions. We prove that the proposed time stepping method has the convergence order $$O(\tau ^{1+ \alpha }), \, \alpha \in (0, 1)$$ O ( τ 1 + α ) , α ∈ ( 0 , 1 ) for general sectorial elliptic operators for nonsmooth data by using the Laplace transform method, where $$\tau $$ τ is the time step size. This convergence order is higher than the convergence orders of the popular convolution quadrature methods (e.g., Lubich’s convolution methods) and L-type methods (e.g., L1 method), which have only $$O(\tau )$$ O ( τ ) convergence for the nonsmooth data. Numerical examples are given to verify the robustness of the time discretization schemes with respect to data regularity.

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.

scholarly journals Analysis of a Decoupled Time-Stepping Scheme for Evolutionary Micropolar Fluid Flows

2016 ◽  
Vol 2016 ◽  
pp. 1-13 ◽  
Author(s):  
S. S. Ravindran
Keyword(s):  
Micropolar Fluid ◽  
Stokes Equations ◽  
Fluid Model ◽  
Navier Stokes ◽  
Optimal Order ◽  
Step Size ◽  
Time Step ◽  
Order Error ◽  
Time Step Size ◽  
Time Stepping

Micropolar fluid model consists of Navier-Stokes equations and microrotational velocity equations describing the dynamics of flows in which microstructure of fluid is important. In this paper, we propose and analyze a decoupled time-stepping algorithm for the evolutionary micropolar flow. The proposed method requires solving only one uncoupled Navier-Stokes and one microrotation subphysics problem per time step. We derive optimal order error estimates in suitable norms without assuming any stability condition or time step size restriction.

Exact Solution of Time History Response for Dynamic Systems With Arbitrary Viscous Damping Using Complex Modal Analysis

2007 ◽  
Author(s):  
Lonny L. Thompson ◽  
Manoj Kumar M. Chinnakonda
Keyword(s):  
Modal Analysis ◽  
State Space ◽  
Piecewise Linear ◽  
Time History ◽  
Quadrature Rules ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Complex Eigenvalues ◽  
Time History Response

A solution method for general, non-proportional damping time history response for piecewise linear loading is generalized to exact solutions which include piecewise quadratic loading. Comparisons are made to Trapezoidal and Simpson’s quadrature rules for approximating the time integral of the weighted generalized forcing function in the exact solution to the decoupled modal equations arising from state-space modal analysis of linear dynamic systems. Closed-form expressions for the weighting parameters in the quadrature formulas in terms of time-step size and complex eigenvalues are derived. The solution is obtained step-by-step from update formulas derived from the piecewise linear and quadratic interpolatory quadrature rules starting from the initial condition. An examination of error estimates for the different force interpolation methods shows convergence rates depend explicitly on the amount of damping in the system as measured by the real-part of the complex eigenvalues of the state-space modal equations and time-step size. Numerical results for a system with general, non-proportional damping, and driven by a continuous loading shows that for systems with light damping, update formulas for standard Trapezoidal and Simpson’s rule integration have comparable accuracy to the weighted piecewise linear and quadratic force interpolation update formulas, while for heavy damping, the update formulas from the weighted force interpolation quadrature rules are more accurate. Using a simple model representing a stiff system with general damping, we show that a two-step modal analysis using real-valued modal reduction followed by state-space modal analysis is shown to be an effective approach for rejecting spurious modes in the spatial discretization of a continuous system.

The Dispersion in One- and Two-Dimensional Finite Element Solutions to the Heat Equation

2002 ◽  
Author(s):  
W. Dauksher ◽  
A. F. Emery
Keyword(s):  
Finite Element ◽  
Heat Equations ◽  
Two Dimensional ◽  
Step Size ◽  
Matrix Formulation ◽  
Time Step ◽  
Time Step Size ◽  
Time Stepping ◽  
Dimensional Finite Element ◽  
The One

The dispersive errors in the finite element solution to the one- and two-dimensional heat equations are examined as a function of element type and size, capacitance matrix formulation, time stepping scheme and time step size.

scholarly journals Energy-corrected FEM and explicit time-stepping for parabolic problems

2019 ◽  
Vol 53 (6) ◽  
pp. 1893-1914
Author(s):  
Piotr Swierczynski ◽  
Barbara Wohlmuth
Keyword(s):  
Finite Element ◽  
Finite Element Approximation ◽  
Regularity Of Solutions ◽  
Parabolic Problems ◽  
Element Approximation ◽  
Computational Domain ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Time Stepping

The presence of corners in the computational domain, in general, reduces the regularity of solutions of parabolic problems and diminishes the convergence properties of the finite element approximation introducing a so-called “pollution effect”. Standard remedies based on mesh refinement around the singular corner result in very restrictive stability requirements on the time-step size when explicit time integration is applied. In this article, we introduce and analyse the energy-corrected finite element method for parabolic problems, which works on quasi-uniform meshes, and, based on it, create fast explicit time discretisation. We illustrate these results with extensive numerical investigations not only confirming the theoretical results but also showing the flexibility of the method, which can be applied in the presence of multiple singular corners and a three-dimensional setting. We also propose a fast explicit time-stepping scheme based on a piecewise cubic energy-corrected discretisation in space completed with mass-lumping techniques and numerically verify its efficiency.

scholarly journals Super-Time-Stepping Scheme for Option Pricing

2020 ◽  
Vol 13 (13) ◽  
pp. 51-54
Author(s):  
Kedar Nath Uprety ◽  
Harithar Khanal ◽  
Ananta Upreti
Keyword(s):  
Option Pricing ◽  
Euler Method ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Time Stepping ◽  
Black Scholes ◽  
Explicit Finite Difference ◽  
The Stability ◽  
Explicit Finite Difference Method

We solve the Black - Scholes equation for option pricing numerically using an Explicit finite difference method. To overcome the stability restriction of the explicit scheme for parabolic partial differential equations in the time step size Courant-Friedrichs-Lewy (CFL) condition, we employ a Super Time Stepping (STS) strategy based on modified Chebyshev polynomial. The numerical results show that the STS scheme boasts of large efficiency gains compared to the standard explicit Euler method.

scholarly journals A decoupled Crank-Nicolson time-stepping scheme for thermally coupled magneto-hydrodynamic system

2017 ◽  
Vol 8 (1) ◽  
pp. 43-62
Author(s):  
S. S. Ravindran
Keyword(s):  
Heat Equation ◽  
Mixed Finite Element ◽  
Mhd Equations ◽  
Optimal Order ◽  
Step Size ◽  
Time Step ◽  
Order Error ◽  
Time Step Size ◽  
Time Stepping ◽  
Crank Nicolson

Thermally coupled magneto-hydrodynamics (MHD) studies the dynamics of electro-magnetically and thermally driven flows,involving MHD equations coupled with heat equation. We introduce a partitioned method that allows one to decouplethe MHD equations from the heat equation at each time step and solve them separately. The extrapolated Crank-Nicolson time-stepping scheme is used for time discretizationwhile mixed finite element method is used for spatial discretization. We derive optimal order error estimates in suitable norms without assuming any stability condition or restrictions on the time step size. We prove the unconditional stability of the scheme. Numerical experiments are used to illustrate the theoretical results.

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.

scholarly journals Adaptive Optimal -Stage Runge-Kutta Methods for Solving Reaction-Diffusion-Chemotaxis Systems

2011 ◽  
Vol 2011 ◽  
pp. 1-25
Author(s):  
Jui-Ling Yu
Keyword(s):  
Method Of Lines ◽  
Reaction Diffusion ◽  
Stability And Convergence ◽  
Optimal Time ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Time Stepping ◽  
Application Problem ◽  
Runge Kutta

We present a class of numerical methods for the reaction-diffusion-chemotaxis system which is significant for biological and chemistry pattern formation problems. To solve reaction-diffusion-chemotaxis systems, efficient and reliable numerical algorithms are essential for pattern generations. Along with the implementation of the method of lines, implicit or semi-implicit schemes are typical time stepping solvers to reduce the effect on time step constrains due to the stability condition. However, these two schemes are usually difficult to employ. In this paper, we propose an adaptive optimal time stepping strategy for the explicit -stage Runge-Kutta method to solve reaction-diffusion-chemotaxis systems. Instead of relying on empirical approaches to control the time step size, variable time step sizes are given explicitly. Yet, theorems about stability and convergence of the algorithm are provided in analyzing robustness and efficiency. Numerical experiment results on a testing problem and a real application problem are shown.

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