# 1.2.1.2. 1D First-order Non-Linear Convection - The Inviscid Burgers’ Equation¶

## 1.2.1.2.1. Understand the Problem¶

• What is the final velocity profile for 1D non-linear convection when the initial conditions are a square wave and the boundary conditions are constant?

• 1D non-linear convection is described as follows:

${\partial u \over \partial t} + u {\partial u \over \partial x} = 0$
• This equation is capable of generating discontinuities (shocks)

## 1.2.1.2.2. Formulate the Problem¶

• Same as Linear Convection

## 1.2.1.2.3. Design Algorithm to Solve Problem¶

### 1.2.1.2.3.1. Space-time discretisation¶

• Same as Linear Convection

### 1.2.1.2.3.2. Numerical scheme¶

• Same as Linear Convection

### 1.2.1.2.3.3. Discrete equation¶

${{u_i^{n+1} - u_i^n} \over {\Delta t}} + c {{u_i^n - u_{i-1}^n} \over \Delta x}=0$

### 1.2.1.2.3.4. Transpose¶

$u_i^{n+1} = u_i^n - c{\Delta t \over \Delta x}(u_i^n - u_{i-1}^n)$

### 1.2.1.2.3.5. Pseudo-code¶

• Very similar to Linear Convection

## 1.2.1.2.4. Implement Algorithm in Python¶

• Very similar to Linear Convection (png, hires.png, pdf) (png, hires.png, pdf) (png, hires.png, pdf)

## 1.2.1.2.5. Conclusions¶

### 1.2.1.2.5.1. Why isn’t the square wave maintained?¶

• The first order backward differencing scheme in space still creates false diffusion as before.

• However, due to the non-linearity in the governing equation, if the spatial step is reduced, the solution can develop shocks, see Figure 2.

• Clearly a square wave is not best represented with the inviscid Burgers Equation.

### 1.2.1.2.5.2. Why does the wave shift to the right?¶

• The square wave is being convected by the velocity, u which is not constant.

• The greatest shift is where the velocity is greatest, see Figure 1

### 1.2.1.2.5.3. What happens at the wall?¶

• As there is no viscosity, there is a non-physical change in the profile near the wall, see Figure 3.

• Comparing this with the linear example, there is clearly much more numerical diffusion in the non-linear example, perhaps due to the convective term being larger, causing a greater magnitude in numerical diffusion.