scholarly journals A hybrid method for solving time fractional advection–diffusion equation on unbounded space domain

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
H. Azin ◽  
F. Mohammadi ◽  
M. H. Heydari
Keyword(s):  
Diffusion Equation ◽  
Hybrid Method ◽  
Hybrid Approach ◽  
Test Problems ◽  
Legendre Functions ◽  
Space Domain ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
Derivative Matrix ◽  
Cardinal Functions

Abstract In this article, a hybrid method is developed for solving the time fractional advection–diffusion equation on an unbounded space domain. More precisely, the Chebyshev cardinal functions are used to approximate the solution of the problem over a bounded time domain, and the modified Legendre functions are utilized to approximate the solution on an unbounded space domain with vanishing boundary conditions. The presented method converts solving this equation into solving a system of algebraic equations by employing the fractional derivative matrix of the Chebyshev cardinal functions and the classical derivative matrix of the modified Legendre functions together with the collocation technique. The accuracy of the presented hybrid approach is investigated on some test problems.

Hybrid Hermite polynomial chaos SBP-SAT technique for stochastic advection-diffusion equations

2020 ◽  
Vol 31 (09) ◽  
pp. 2050128
Author(s):  
Navjot Kaur ◽  
Kavita Goyal
Keyword(s):  
Diffusion Equation ◽  
Hermite Polynomial ◽  
Polynomial Chaos ◽  
Real Life ◽  
Computational Effort ◽  
Dirichlet Boundary Conditions ◽  
Diffusion Equations ◽  
Test Problems ◽  
Advection Diffusion Equation ◽  
Advection Diffusion

The study of advection–diffusion equation has relatively became an active research topic in the field of uncertainty quantification (UQ) due to its numerous real life applications. In this paper, Hermite polynomial chaos is united with summation-by-parts (SBP) – simultaneous approximation terms (SATs) technique to solve the advection–diffusion equations with random Dirichlet boundary conditions (BCs). Polynomial chaos expansion (PCE) with Hermite basis is employed to separate the randomness, then SBP operators are used to approximate the differential operators and SATs are used to enforce BCs by ensuring the stability. For each chaos coefficient, time integration is performed using Runge–Kutta method of fourth order (RK4). Statistical moments namely mean and variance are computed using polynomial chaos coefficients without any extra computational effort. The method is applied on three test problems for validation. The first two test problems are stochastic advection equations on [Formula: see text] without any boundary and third problem is stochastic advection–diffusion equation on [0,2] with Dirichlet BCs. In case of third problem, we have obtained a range of permissible parameters for a stable numerical solution.

A POD-Based Solver for the Advection-Diffusion Equation

2011 ◽  
Author(s):  
Elia Merzari ◽  
W. David Pointer ◽  
Paul Fischer
Keyword(s):  
Diffusion Equation ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Galerkin Projection ◽  
Test Problems ◽  
Simulation Code ◽  
Dimensional Flow ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
Two Dimensional Flow

We present a methodology based on proper orthogonal decomposition (POD). We have implemented the POD-based solver in the large eddy simulation code Nek5000 and used it to solve the advection-diffusion equation for temperature in cases where buoyancy is not present. POD allows for the identification of the most energetic modes of turbulence when applied to a sufficient set of snapshots generated through Nek5000. The Navier-Stokes equations are then reduced to a set of ordinary differential equations by Galerkin projection. The flow field is reconstructed and used to advect the temperature on longer time scales and potentially coarser grids. The methodology is validated and tested on two problems: two-dimensional flow past a cylinder and three-dimensional flow in T-junctions. For the latter case, the benchmark chosen corresponds to the experiments of Hirota et al., who performed particle image velocimetry on the flow in a counterflow T-junction. In both test problems the dynamics of the reduced-order model reproduce well the history of the projected modes when a sufficient number of equations are considered. The dynamics of flow evolution and the interaction of different modes are also studied in detail for the T-junction.

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.

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