Convection Diffusion

Consider the convection-diffusion equation

$\large \frac{\partial T}{\partial t} + u \frac{\partial T}{\partial x} = \alpha \frac{\partial^2 T}{\partial x^2}$

With the boundary conditions

Part 1. Pure convection (α = 0)

Consider the following initial profile

$\large T(x,0)=\left\{\begin{matrix} 1-(10x-1)^2 & at \; \: 0<x\leq 0.2 \\ 0 & at \; \: 0.2<x\leq 1 \end{matrix}\right.$

The exact solution is

$\large T(x,t)=\left\{\begin{matrix} 1-(10(x-ut)-1)^2 & at \; \: 0<x-ut\leq 0.2 \\ 0 & at \; \: 0.2<x-ut\leq 1 \end{matrix}\right.$

(1) Explicit Euler time advancement and second-order central difference for the spatial derivative.

Discretization:

$\large \frac{T^{n+1}_{j}-T^{n}_{j}}{\Delta t} + u \frac{T^{n}_{j+1}-T^{n}_{j-1}}{2\Delta x} = 0$

From modified wave number analysis, Explicit Euler is unstable.

(2) Leapfrog time advancement and the second-order central difference for the spatial derivative.

Discretization:

$\large \frac{T^{n+1}_{j}-T^{n-1}_{j}}{2\Delta t} + u \frac{T^{n}_{j+1}-T^{n}_{j-1}}{2\Delta x} = 0$

From modified wave number analysis, Leapfrog is stable when

$\large \Delta t \leq 12.5 \Delta x$

Part 2. Convection-diffusion

Let $\alpha = 0.001$

(1) Explicit Euler time advancement and second-order central difference for the spatial derivative.

Discretization:

$\large \frac{T^{n+1}_{j}-T^{n}_{j}}{\Delta t} + u \frac{T^{n}_{j+1}-T^{n}_{j-1}}{2\Delta x} = \alpha \frac{T^{n}_{j+1}-2T^{n}_{j}+T^{n}_{j-1}}{\Delta x^2}$

From Von-Neumann stability analysis, Explicit Euler is stable when

$\large \begin{matrix} \Delta t \leq 500 \Delta x^2 \\ \Delta t \leq 12.5 \Delta x \end{matrix}$

(2) Leapfrog time advancement and the second-order central difference for the spatial derivative.

Discretization:

$\large \frac{T^{n+1}_{j}-T^{n-1}_{j}}{2\Delta t} + u \frac{T^{n}_{j+1}-T^{n}_{j-1}}{2\Delta x} = \alpha \frac{T^{n}_{j+1}-2T^{n}_{j}+T^{n}_{j-1}}{\Delta x^2}$

From modified wave number analysis, Leapfrog is unstable.

Simulation Result

Let $\large u=0.08$ , $\large \Delta t=0.05$ ,and $\large \Delta x=0.01$

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README.md

README.md

Convection Diffusion

Part 1. Pure convection (α = 0)

(1) Explicit Euler time advancement and second-order central difference for the spatial derivative.

(2) Leapfrog time advancement and the second-order central difference for the spatial derivative.

Part 2. Convection-diffusion

(1) Explicit Euler time advancement and second-order central difference for the spatial derivative.

(2) Leapfrog time advancement and the second-order central difference for the spatial derivative.

Simulation Result

Part 1. Pure convection (α = 0)

Part 2. Convection-diffusion

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Convection Diffusion

Part 1. Pure convection (α = 0)

(1) Explicit Euler time advancement and second-order central difference for the spatial derivative.

(2) Leapfrog time advancement and the second-order central difference for the spatial derivative.

Part 2. Convection-diffusion

(1) Explicit Euler time advancement and second-order central difference for the spatial derivative.

(2) Leapfrog time advancement and the second-order central difference for the spatial derivative.

Simulation Result

Part 1. Pure convection (α = 0)

Part 2. Convection-diffusion