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

1.2.1.2.5. 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.