1D First-order Non-Linear Convection - The Inviscid Burgers’ Equation 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) Formulate the Problem

  • Same as Linear Convection Design Algorithm to Solve Problem Space-time discretisation

  • Same as Linear Convection Numerical scheme

  • Same as Linear Convection Discrete equation

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

\[u_i^{n+1} = u_i^n - c{\Delta t \over \Delta x}(u_i^n - u_{i-1}^n)\] Pseudo-code

  • Very similar to Linear Convection Conclusions 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. 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 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.