Intermediate dirichlet boundary conditions for operator splitting algorithms for the advection-diffusion equation

1995 ◽  
Vol 24 (4) ◽  
pp. 447-458 ◽  
Author(s):  
Liaqat Ali Khan ◽  
Philip L.-F. Liu
Keyword(s):  
Diffusion Equation ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Operator Splitting ◽  
Dirichlet Boundary Conditions ◽  
Splitting Algorithms ◽  
Advection Diffusion Equation ◽  
Advection Diffusion

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.

scholarly journals Explicit finite difference solution for contaminant transport problems with constant and oscillating boundary conditions

2020 ◽  
Vol 24 (3 Part B) ◽  
pp. 2225-2231
Author(s):  
Svetislav Savovic ◽  
Alexandar Djordjevich
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Diffusion Equation ◽  
Boundary Conditions ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
Oscillating Boundary ◽  
Constant Coefficients ◽  
Explicit Finite Difference ◽  
Explicit Finite Difference Method

For constant and oscillating boundary conditions, the 1-D advection-diffusion equation with constant coefficients, which describes a contaminant flow, is solved by the explicit finite difference method in a semi-infinite medium. It is shown how far the periodicity of the oscillating boundary carries on until diminishing to below appreciable levels a specified distance away, which depends on the oscillation characteristics of the source. Results are tested against an analytical solution reported for a special case. The explicit finite difference method is shown to be effective for solving the advection-diffusion equation with constant coefficients in semi-infinite media with constant and oscillating boundary conditions.

scholarly journals Explicit finite-difference solution of two-dimensional solute transport with periodic flow in homogenous porous media

2017 ◽  
Vol 65 (4) ◽  
pp. 426-432 ◽  
Author(s):  
Alexandar Djordjevich ◽  
Svetislav Savović ◽  
Aco Janićijević
Keyword(s):  
Finite Difference ◽  
Diffusion Equation ◽  
Boundary Conditions ◽  
Variable Coefficients ◽  
Two Dimensional ◽  
Seepage Velocity ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
Special Cases ◽  
Explicit Finite Difference

AbstractThe two-dimensional advection-diffusion equation with variable coefficients is solved by the explicit finitedifference method for the transport of solutes through a homogenous two-dimensional domain that is finite and porous. Retardation by adsorption, periodic seepage velocity, and a dispersion coefficient proportional to this velocity are permitted. The transport is from a pulse-type point source (that ceases after a period of activity). Included are the firstorder decay and zero-order production parameters proportional to the seepage velocity, and periodic boundary conditions at the origin and at the end of the domain. Results agree well with analytical solutions that were reported in the literature for special cases. It is shown that the solute concentration profile is influenced strongly by periodic velocity fluctuations. Solutions for a variety of combinations of unsteadiness of the coefficients in the advection-diffusion equation are obtainable as particular cases of the one demonstrated here. This further attests to the effectiveness of the explicit finite difference method for solving two-dimensional advection-diffusion equation with variable coefficients in finite media, which is especially important when arbitrary initial and boundary conditions are required.

FRACTIONAL STEP METHODS FOR SPECTRAL APPROXIMATION OF ADVECTION-DIFFUSION EQUATIONS

1996 ◽  
Vol 06 (07) ◽  
pp. 1027-1050 ◽  
Author(s):  
PAOLA GERVASIO
Keyword(s):  
Diffusion Equation ◽  
Boundary Conditions ◽  
Peclet Number ◽  
Diffusion Equations ◽  
Fractional Step ◽  
Spectral Approximation ◽  
Two Dimensional ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
Large Peclet Number

Several classical fractional step schemes are proposed for the spectral approximation of advection-diffusion equation in two-dimensional geometries. Suitable boundary conditions are studied in order to preserve the accuracy of the schemes at each step. An example is given for the application of the fractional step schemes to solve problems with large Péclet number.

High-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations (II)

2015 ◽  
Vol 18 (3) ◽  
Author(s):  
Jianxiong Cao ◽  
Changpin Li ◽  
YangQuan Chen
Keyword(s):  
Dirichlet Boundary ◽  
Order Approximation ◽  
High Order ◽  
Dirichlet Boundary Conditions ◽  
High Order Accuracy ◽  
High Order Approximation ◽  
Order Accuracy ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
New Formula

AbstractIn this paper, we first establish a high-order numerical algorithm for α-th (0 < α < 1) order Caputo derivative of a given function f(t), where the convergence rate is (4 − α)-th order. Then by using this new formula, an improved difference scheme with high order accuracy in time to solve Caputo-type fractional advection-diffusion equation with Dirichlet boundary conditions is constructed. Finally, numerical examples are carried out to confirm the efficiency of the constructed algorithm.

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