View source for Numerical solution of the convection–diffusion equation

The [[convection–diffusion equation]] describes the flow of heat, particles, or other physical quantities in situations where there is both [[diffusion]] and [[convection]] or [[advection]]. For information about the equation, its derivation, and its conceptual importance and consequences, see the main article [[convection–diffusion equation]]. This article describes how to use a computer to calculate an approximate numerical solution of the discretized equation, in a time-dependent situation.

In order to be concrete, this article focuses on ''heat flow'', an important example where the convection–diffusion equation applies. However, the same mathematical analysis works equally well to other situations like particle flow.

A general discontinuous [[finite element]] formulation is needed.<ref>“Discontinuous Finite in Fluid Dynamics and [[Heat transfer]]” by Ben Q. Li, 2006.</ref> The unsteady convection–diffusion problem is considered, at first the known temperature T is expanded into a [[Taylor series]] with respect to time taking into account its three components. Next, using the convection diffusion equation an equation is obtained from the [[Differentiation (mathematics)|differentiation]] of this equation.

== Equation ==

=== General ===

The following convection diffusion equation is considered here<ref>"The [[Finite Difference Method]] For Transient Convection Diffusion", Ewa Majchrzak & Łukasz Turchan, 2012.</ref>

<math display="block">c \rho\left[\frac{\partial T(x,t)}{\partial t} + \epsilon u \frac{\partial T(x,t)}{\partial x}\right]=\lambda \frac{\partial^2 T(x,t)}{\partial x^2} + Q(x,t)</math>

In the above equation, four terms represents [[transient state (chemical engineering)|transience]], [[convection]], [[diffusion]] and a source term respectively, where
* {{mvar|T}} is the temperature in particular case of [[heat transfer]] otherwise it is the variable of interest
* {{mvar|t}} is time
* {{mvar|c}} is the specific heat
* {{mvar|u}} is velocity
* {{mvar|ε}} is porosity that is the ratio of liquid volume to the total volume
* {{mvar|ρ}} is mass density
* {{mvar|λ}} is thermal conductivity
* {{math|''Q''(''x'',''t'')}} is source term representing the capacity of internal sources

The equation above can be written in the form

<math display="block">\frac{\partial T}{\partial t} = a \frac{\partial^2 T}{\partial x^2} - \epsilon u \frac{\partial T}{\partial x} + \frac{Q}{c \rho}</math>

where {{math|''a'' {{=}} {{sfrac|''λ''|''cρ''}}}} is the diffusion coefficient.

=== Solving the convection–diffusion equation using the finite difference method ===

A solution of the transient convection–diffusion equation can be approximated through a [[finite difference]] approach, known as the [[finite difference method]] (FDM).

==== Explicit scheme ====

An explicit scheme of FDM has been considered and stability criteria are formulated. In this scheme, temperature is totally dependent on the old temperature (the initial conditions) and {{mvar|θ}}, a weighting parameter between 0 and 1. Substitution of {{math|''θ'' {{=}} 0}} gives the explicit [[discretization]] of the unsteady conductive heat transfer equation.

<math display="block">\frac{T_i ^f - T_i ^{f-1}}{\Delta t} = a \frac{T_{i-1} ^{f-1} - 2T_i ^{f-1} + T_{i+1} ^{f-1}}{h^2} - \epsilon u \frac{T_{i+1} ^{f-1} - T_{i-1} ^{f-1}}{2h} + \frac{Q_i ^{f-1}}{c \rho}</math>

where
* {{math|Δ''t'' {{=}} ''t''<sup>''f''</sup> − ''t''<sup>''f'' − 1</sup>}}
* {{mvar|h}} is the uniform grid spacing (mesh step)

<math display="block">T_i^f = \left(1-\frac{2a\Delta t}{h^2}\right) T_i ^{f-1} + \left(\frac{a \Delta t}{h^2} + \frac{\epsilon u \Delta t}{2h}\right) T_{i-1} ^{f-1} + \left(\frac{a \Delta t}{h^2} - \frac{\epsilon u \Delta t}{2h}\right) T_{i+1} ^{f-1} + \frac{Q_i ^{f-1}}{c \rho} \Delta t</math>

===== Stability criteria =====
The inequalities
<math display="block">\begin{align}
h &< \frac{2a}{|\epsilon u|}, &
\Delta t &<\frac a{\epsilon^2u^2/4+a^2/h^2} <\frac{h^2}{2a}<\frac{h}{|\epsilon u|}
\end{align}</math>
follow from setting <math>Q^f_i\to0</math> and requiring the [[ansatz]] <math>T^f_i\to T^f_0\exp(\sqrt{-1}i\theta)</math> not to gain amplitude as {{mvar|f}} increases for any <math>\theta</math>.
They set a stringent maximum limit to the time step that represents a serious limitation for the explicit scheme. This method is not recommended for general transient problems because the maximum possible time step has to be reduced as the square of {{mvar|h}}.

==== Implicit scheme ====
In implicit scheme, the temperature is dependent at the new time level {{math|''t'' + Δ''t''}}. After using implicit scheme, it was found that all coefficients are positive. It makes the implicit scheme unconditionally stable for any size of time step. This scheme is preferred for general purpose transient calculations because of its robustness and unconditional stability.<ref>H.Versteeg & W. Malalasekra, "an Introduction to [[Computational Fluid Dynamics]]" 2009, pages 262–263.</ref> The disadvantage of this method is that more procedures are involved and due to larger {{math|Δ''t''}}, truncation error is also larger.

==== Crank–Nicolson scheme ====

In the [[Crank–Nicolson method]], the temperature is equally dependent on {{mvar|t}} and {{math|''t'' + Δ''t''}}. It is a second-[[Orders of approximation|order]] method in time and this method is generally used in [[diffusion]] problems.

===== Stability criteria =====

<math display="block">\Delta t < \frac{h^2}{a}</math>

This time step limitation is less restricted than the [[explicit method]]. The [[Crank–Nicolson method]] is based on the central differencing and hence it is second-order accurate in time.<ref>H.Versteeg & W. Malalasekra, "an Introduction to [[Computational Fluid Dynamics]]" 2009, page no. 262.</ref>

== Finite element solution to convection–diffusion problem ==

Unlike the conduction equation (a finite element solution is used), a numerical solution for the [[convection–diffusion equation]] has to deal with the convection part of the governing equation in addition to diffusion. When the [[Péclet number]] (Pe) exceeds a critical value, the spurious oscillations result in space and this problem is not unique to finite elements as all other [[discretization]] techniques have the same difficulties. In a finite difference formulation, the spatial oscillations are reduced by a family of discretization schemes like [[upwind scheme]].<ref>Ronald W. Lewis, Perumal Nithiarasu & Kankanhally N. Seetharamu, "Fundamentals for the [[finite element method]] for heat and fluid flow".</ref> In this method, the basic shape function is modified to obtain the upwinding effect. This method is an extension of [[Runge–Kutta]] discontinuous for a convection-diffusion equation.
For time-dependent equations, a different kind of approach is followed. The [[finite difference scheme]] has an equivalent in the [[finite element method]] ([[Galerkin method]]). Another similar method is the characteristic Galerkin method (which uses an implicit algorithm). For scalar variables, the above two methods are identical.

== See also ==
* [[Advanced Simulation Library]]
* [[Convection–diffusion equation]]
* [[Double diffusive convection]]
* [[An Album of Fluid Motion]]
* [[Lagrangian and Eulerian specification of the flow field]]
* [[Fluid animation]]
* [[Finite volume method for unsteady flow]]

== References ==
{{Reflist}}

{{DEFAULTSORT:Numerical solution of the convection-diffusion equation}}
[[Category:Diffusion]]
[[Category:Parabolic partial differential equations]]
[[Category:Stochastic differential equations]]
[[Category:Transport phenomena]]
[[Category:Equations of physics]]