Numerical solution of burgers equation with finite volume method and crank-nicolson

Question

I'm having difficulty with numerically solving the inviscid burgers equation.Godunov's scheme is used in most of what I've found in literature . Now my question is if using a crank nicolson shceme is wrong or not?

$\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x}=0$

with this BC:

at $t> 0 ,x=0$ : $u=1$

using a finite volume method:

$\frac{\partial }{\partial t}\int udv + \int u\frac{\partial u}{\partial x}dv=0 $

$\frac{\partial u}{\partial t}\Delta v +(uuA) _{e}-(uuA) _{w}=0$

$\frac{\partial u}{\partial t} + F_{e}u_{e}-F_{w}u_{w}=0 $

$F=\frac{u}{\Delta x}$

and for discretization in time I used Crank-Nicolson

$u_{p}^{n+1}-u_{p}^{n}=\frac{\Delta t}{2}(-F_{e}u_{e}+F_{w}u_{w})^{n+1}+\frac{\Delta t}{2}(-F_{e}u_{e}+F_{w}u_{w})^{n}$

the final form of equation is as (upwind scheme) :

$(1+\frac{\Delta t}{2}F_{e})u_{P}^{n+1}=(-\frac{\Delta t}{2}F_{w}) u_{W}^{n+1} +u_{P}^{n}+\frac{\Delta t}{2}(F_{w}u_{w}-F_{e}u_{e})^{n} $

and for the first block :

$(1+\frac{\Delta t}{2}F_{e})u_{P}^{n+1}= u_{P}^{n}+\frac{\Delta t}{2}(-F_{e}u_{e})^{n}+\Delta t F_{w}u_{w}$

and after that I tried to solve a set of linear algebraic equation. The result in every time step seems to converge but with time passing the magnitude of velocity is also increasing which is an indication of a mistake.

Answer 1

If I understand correctly, you are using a centered finite difference in space and the implicit trapezoidal method in time. That scheme is unconditionally absolutely stable, but will generate spurious oscillations. So you should expect to see some increase in the maximum value of $u$, but it shouldn't blow up. If it blows up, you have an implementation error (bug).

I'll add that this is a lousy method for a purely hyperbolic PDE, since it generates oscillations and isn't very accurate for large time steps anyway.

Answer 2

Thank you for your clarification.

The Thomas algorithm solves $Ax=b$ when, e.g., A is a tridiagonal matrix (there are other special cases I believe, but this is not one of them). Your "final form of equation" does not appear to be in this form, nor does it look like this is possible.

As @David Ketcheson mentioned, implicit time marching is not very attractive for problems like this. Usually the diffusion term is treated implicitly in time, and for good reason.

Explicit time marching has numerical stability restrictions due to the advection and diffusion terms

$\Delta t < C_1 \frac{\Delta x}{\max(U)}$

$\Delta t < C_2 \frac{\Delta x^2}{\nu}$

respectively. The latter is often more restrictive (for diffusion dominant problems or when node points are highly clustered). Often, the advection term is treated explicitly and the CFL condition is used to determine the largest allowable time step when the diffusion term is not a problem.

As @David Ketcheson mentioned in his comment, explicit Euler is not a good choice for central differencing since it is unconditionally unstable. However, upwinding schemes with explicit time marching is frequently used in high Reynolds number CFD, so I imagine that it is applicable here, however, you cannot use central differencing since you have no diffusion.

As for spatial discretization, there is a lot of literature on upwinding schemes.