scholarly journals Error analysis of nonlinear time fractional mobile/immobile advection-diffusion equation with weakly singular solutions

2021 ◽  
Vol 24 (1) ◽  
pp. 202-224
Author(s):  
Hui Zhang ◽  
Xiaoyun Jiang ◽  
Fawang Liu
Keyword(s):  
Computational Cost ◽  
Correction Method ◽  
Trapezoidal Rule ◽  
Stability And Convergence ◽  
Fast Method ◽  
Memory Requirement ◽  
Time Step ◽  
Advection Dispersion ◽  
Advection Diffusion Equation ◽  
The Stability

Abstract In this paper, a weighted and shifted Grünwald-Letnikov difference (WSGD) Legendre spectral method is proposed to solve the two-dimensional nonlinear time fractional mobile/immobile advection-dispersion equation. We introduce the correction method to deal with the singularity in time, and the stability and convergence analysis are proven. In the numerical implementation, a fast method is applied based on a globally uniform approximation of the trapezoidal rule for the integral on the real line to decrease the memory requirement and computational cost. The memory requirement and computational cost are O(Q) and O(QK), respectively, where K is the number of the final time step and Q is the number of quadrature points used in the trapezoidal rule. Some numerical experiments are given to confirm our theoretical analysis and the effectiveness of the presented methods.

A Robust Implementation of a Reynolds Stress Model for Turbomachinery Applications in a Coupled Solver Environment

2020 ◽  
Author(s):  
David Roos Launchbury ◽  
Luca Mangani ◽  
Ernesto Casartelli ◽  
Francesco Del Citto
Keyword(s):  
Reynolds Stress ◽  
Turbulence Modeling ◽  
Computational Cost ◽  
Reynolds Stress Model ◽  
Tip Vortex ◽  
Stability And Convergence ◽  
Stress Model ◽  
Robust Implementation ◽  
Flow Phenomena ◽  
The Stability

Abstract In the industrial simulation of flow phenomena, turbulence modeling is of prime importance. Due to their low computational cost, Reynolds-averaged methods (RANS) are predominantly used for this purpose. However, eddy viscosity RANS models are often unable to adequately capture important flow physics, specifically when strongly anisotropic turbulence and vortex structures are present. In such cases the more costly 7-equation Reynolds stress models often lead to significantly better results. Unfortunately, these models are not widely used in the industry. The reason for this is not mainly the increased computational cost, but the stability and convergence issues such models usually exhibit. In this paper we present a robust implementation of a Reynolds stress model that is solved in a coupled manner, increasing stability and convergence speed significantly compared to segregated implementations. In addition, the decoupling of the velocity and Reynolds stress fields is addressed for the coupled equation formulation. A special wall function is presented that conserves the anisotropic properties of the model near the walls on coarser meshes. The presented Reynolds stress model is validated on a series of semi-academic test cases and then applied to two industrially relevant situations, namely the tip vortex of a NACA0012 profile and the Aachen Radiver radial compressor case.

A ROBUST IMPLEMENTATION OF A REYNOLDS STRESS MODEL FOR TURBOMACHINERY APPLICATIONS IN A COUPLED SOLVER ENVIRONMENT

2021 ◽  
pp. 1-31
Author(s):  
David Roos Launchbury ◽  
Luca Mangani ◽  
Ernesto Casartelli ◽  
Francesco Del Citto
Keyword(s):  
Reynolds Stress ◽  
Turbulence Modeling ◽  
Computational Cost ◽  
Reynolds Stress Model ◽  
Tip Vortex ◽  
Stability And Convergence ◽  
Stress Model ◽  
Robust Implementation ◽  
Flow Phenomena ◽  
The Stability

Abstract In the industrial simulation of flow phenomena, turbulence modeling is of prime importance. Due to their low computational cost, Reynolds-averaged methods (RANS) are predominantly used for this purpose. However, eddy viscosity RANS models are often unable to adequately capture important flow physics, specifically when strongly anisotropic turbulence and vortex structures are present. In such cases the more costly 7-equation Reynolds stress models often lead to significantly better results. Unfortunately, these models are not widely used in the industry. The reason for this is not mainly the increased computational cost, but the stability and convergence issues such models usually exhibit. In this paper we present a robust implementation of a Reynolds stress model that is solved in a coupled manner, increasing stability and convergence speed significantly compared to segregated implementations. In addition, the decoupling of the velocity and Reynolds stress fields is addressed for the coupled equation formulation. A special wall function is presented that conserves the anisotropic properties of the model near the walls on coarser meshes. The presented Reynolds stress model is validated on a series of semi-academic test cases and then applied to two industrially relevant situations, namely the tip vortex of a NACA0012 profile and the Aachen Radiver radial compressor case.

scholarly journals Novel Second-Order Accurate Implicit Numerical Methods for the Riesz Space Distributed-Order Advection-Dispersion Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-14 ◽  
Author(s):  
X. Wang ◽  
F. Liu ◽  
X. Chen
Keyword(s):  
Numerical Methods ◽  
Quadrature Rule ◽  
Second Order ◽  
Riesz Space ◽  
Stability And Convergence ◽  
Dispersion Equations ◽  
Advection Dispersion ◽  
Alternating Direction ◽  
The Stability ◽  
Distributed Order

We derive and analyze second-order accurate implicit numerical methods for the Riesz space distributed-order advection-dispersion equations (RSDO-ADE) in one-dimensional (1D) and two-dimensional (2D) cases, respectively. Firstly, we discretize the Riesz space distributed-order advection-dispersion equations into multiterm Riesz space fractional advection-dispersion equations (MT-RSDO-ADE) by using the midpoint quadrature rule. Secondly, we propose a second-order accurate implicit numerical method for the MT-RSDO-ADE. Thirdly, stability and convergence are discussed. We investigate the numerical solution and analysis of the RSDO-ADE in 1D case. Then we discuss the RSDO-ADE in 2D case. For 2D case, we propose a new second-order accurate implicit alternating direction method, and the stability and convergence of this method are proved. Finally, numerical results are presented to support our theoretical analysis.

scholarly journals Time-step limits for stable solutions of the ice-sheet equation

1996 ◽  
Vol 23 ◽  
pp. 74-85 ◽  
Author(s):  
Richard C. A. Hindmarsh ◽  
Antony J. Payne
Keyword(s):  
Linear Equations ◽  
Computational Cost ◽  
Ice Sheet ◽  
Two Dimensions ◽  
Picard Iteration ◽  
Time Step ◽  
Iterated Maps ◽  
A New Technique ◽  
The Stability ◽  
Non Linear Equations

Various spatial discretizations for the ice sheet are compared for accuracy against analytical solutions in one and two dimensions. The computational efficiency of various iterated and non-iterated marching schemes is compared.The stability properties of different marching schemes, with and without iterations on the non-linear equations, are compared. Newton–Raphson techniques permit the largest time steps. A new technique, which is based on the fact that the dynamics of unstable iterated maps contain information about where the unstable root lies, is shown to improve substantially the performance of Picard iteration at a negligible computational cost.

scholarly journals Time-step limits for stable solutions of the ice-sheet equation

1996 ◽  
Vol 23 ◽  
pp. 74-85 ◽  
Author(s):  
Richard C. A. Hindmarsh ◽  
Antony J. Payne
Keyword(s):  
Linear Equations ◽  
Computational Cost ◽  
Ice Sheet ◽  
Two Dimensions ◽  
Picard Iteration ◽  
Time Step ◽  
Iterated Maps ◽  
A New Technique ◽  
The Stability ◽  
Non Linear Equations

Various spatial discretizations for the ice sheet are compared for accuracy against analytical solutions in one and two dimensions. The computational efficiency of various iterated and non-iterated marching schemes is compared. The stability properties of different marching schemes, with and without iterations on the non-linear equations, are compared. Newton–Raphson techniques permit the largest time steps. A new technique, which is based on the fact that the dynamics of unstable iterated maps contain information about where the unstable root lies, is shown to improve substantially the performance of Picard iteration at a negligible computational cost.

Stability and Convergence of Implicit Numerical Methods for a Class of Fractional Advection-Dispersion Models

2011 ◽  
Author(s):  
Fawang Liu ◽  
Pinghui Zhuang ◽  
Kevin Burrage
Keyword(s):  
Numerical Methods ◽  
Heavy Tails ◽  
Regional Scale ◽  
Wave Model ◽  
Anomalous Dispersion ◽  
Stability And Convergence ◽  
Dispersion Models ◽  
Diffusion Wave ◽  
Advection Dispersion ◽  
The Stability

In this paper, a class of fractional advection-dispersion models (FADM) is investigated. These models include five fractional advection-dispersion models: the immobile, mobile/immobile time FADM with a temporal fractional derivative 0 < γ < 1, the space FADM with skewness, both the time and space FADM and the time fractional advection-diffusion-wave model with damping with index 1 < γ < 2. They describe nonlocal dependence on either time or space, or both, to explain the development of anomalous dispersion. These equations can be used to simulate regional-scale anomalous dispersion with heavy tails, for example, the solute transport in watershed catchments and rivers. We propose computationally effective implicit numerical methods for these FADM. The stability and convergence of the implicit numerical methods are analyzed and compared systematically. Finally, some results are given to demonstrate the effectiveness of our theoretical analysis.

scholarly journals Computationally Efficient Hybrid Method for the Numerical Solution of the 2D Time Fractional Advection-Diffusion Equation

2020 ◽  
Vol 5 (3) ◽  
pp. 432-446
Author(s):  
Fouad Mohammad Salama ◽  
Norhashidah Hj. Mohd Ali
Keyword(s):  
Finite Difference ◽  
Diffusion Equation ◽  
Difference Scheme ◽  
Numerical Solution ◽  
Hybrid Method ◽  
Finite Difference Scheme ◽  
Computational Cost ◽  
Stability And Convergence ◽  
Advection Diffusion Equation ◽  
Advection Diffusion

In this paper, a hybrid method based on the Laplace transform and implicit finite difference scheme is applied to obtain the numerical solution of the two-dimensional time fractional advection-diffusion equation (2D-TFADE). Some of the major limitations in computing the numerical solution for fractional differential equations (FDEs) in multi-dimensional space are the huge computational cost and storage requirement, which are O(N^2) cost and O(MN) storage, N and M are the total number of time levels and space grid points, respectively. The proposed method reduced the computational complexity efficiently as it requires only O(N) computational cost and O(M) storage with reasonable accuracy when numerically solving the TFADE. The method’s stability and convergence are also investigated. The Results of numerical experiments of the proposed method are obtained and found to compare well with the results of existing standard finite difference scheme.

scholarly journals Stability analysis of finite difference schemes for an advection diffusion equation

2017 ◽  
Vol 29 (2) ◽  
pp. 143-151 ◽  
Author(s):  
TMAK Azad ◽  
LS Andallah
Keyword(s):  
Stability Analysis ◽  
Finite Difference ◽  
Diffusion Equation ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Time Step ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
The Stability ◽  
Stability Results

The paper studies stability analysis for two standard finite difference schemes FTBSCS (forward time backward space and centered space) and FTCS (forward time and centered space). One-dimensional advection diffusion equation is solved by using the schemes with appropriate initial and boundary conditions. Numerical experiments are performed to verify the stability results obtained in this study. It is found that FTCS scheme gives better point-wise solutions than FTBSCS in terms of time step selection.Bangladesh J. Sci. Res. 29(2): 143-151, December-2016

The Stability and Convergence of Fully Discrete Galerkin-Galerkin FEMs for Porous Medium Flows

2014 ◽  
Vol 15 (4) ◽  
pp. 1141-1158 ◽  
Author(s):  
Buyang Li ◽  
Jilu Wang ◽  
Weiwei Sun
Keyword(s):  
Theoretical Analysis ◽  
Error Estimates ◽  
Three Dimensional ◽  
Time Discretization ◽  
Stability And Convergence ◽  
Time Step ◽  
Element Discretization ◽  
Flow Models ◽  
Fully Discrete ◽  
The Stability

AbstractThe paper is concerned with the unconditional stability and error estimates of fully discrete Galerkin-Galerkin FEMs for the equations of incompressible miscible flows in porous media. We prove that the optimal L2 error estimates hold without any time-step (convergence) conditions, while all previous works require certain time-step restrictions. Theoretical analysis is based on a splitting of the error into two parts: the error from the time discretization of the PDEs and the error from the finite element discretization of the corresponding time-discrete PDEs, which was proposed in our previous work [26, 27]. Numerical results for both two and three-dimensional flow models are presented to confirm our theoretical analysis.

A Time Splitting Space Spectral Element Method for the Cahn-Hilliard Equation

2013 ◽  
Vol 3 (4) ◽  
pp. 333-351 ◽  
Author(s):  
Lizhen Chen ◽  
Chuanju Xu
Keyword(s):  
Lower Order ◽  
Neumann Boundary ◽  
Mass Conservation ◽  
Spectral Element ◽  
Stability And Convergence ◽  
Time Step ◽  
Fully Discrete ◽  
Hilliard Equation ◽  
Cahn Hilliard Equation ◽  
The Stability

AbstractWe propose and analyse a class of fully discrete schemes for the Cahn-Hilliard equation with Neumann boundary conditions. The schemes combine large-time step splitting methods in time and spectral element methods in space. We are particularly interested in analysing a class of methods that split the original Cahn-Hilliard equation into lower order equations. These lower order equations are simpler and less computationally expensive to treat. For the first-order splitting scheme, the stability and convergence properties are investigated based on an energy method. It is proven that both semi-discrete and fully discrete solutions satisfy the energy dissipation and mass conservation properties hidden in the associated continuous problem. A rigorous error estimate, together with numerical confirmation, is provided. Although not yet rigorously proven, higher-order schemes are also constructed and tested by a series of numerical examples. Finally, the proposed schemes are applied to the phase field simulation in a complex domain, and some interesting simulation results are obtained.

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