# Why do we have to resort to Higher order schemes for solving the 1-D advection equation/ continuity equation?

\begin{equation} \begin{aligned} \frac{\partial N}{\partial t} &+ \frac{\partial J}{\partial r} = 0, \\ \frac{\partial N}{\partial t} &+ \frac{\partial }{\partial r}(N \upsilon )= 0, \end{aligned} \end{equation}

where $$v$$ is varying across the $$r$$ axis.

For solving this equation, do I need the higher order schemes (for e.g. MUSCL employing flux limitor functions) to discretize this instead of FOU? I know that FOU can produce numerical diffusion and cause inaccuracies. If I use FOU (owing to its simplicity), what do I need to keep in mind? Smaller grid size considering grid Peclet number?

Also, I am using FVM method on a non-uniform grid. Is there any good book/resource which can describe this a little further?

FOU : First Order Upwind
MUSCL: Monotone Upstream-centered schemes for Conservative laws
FVM : Finite volume method

• Note: I had to scour the internet to learn people abbreviate first-order upwind as FOU – Spencer Bryngelson May 7 at 5:01
• Your equations do not look right to me. Anyway, we use MUSCL when there are discontinuities in the solution space. The usual Godunov scheme will give you just second-order spatial accuracy. Its advantage is in suppressing spurious oscillations near those discontinuities. For smooth solutions, regular high-order finite differences are (usually) fine. – Spencer Bryngelson May 7 at 5:10
• @SpencerBryngelson, could you point out the errors? And thanks for the clarification. Can you recommend any resources to better understand this? Sorry for not expanding the FOU abbrev. Also, do you think I need higher order schemes for solving this? – Ronnie1993 May 7 at 6:35

## 1 Answer

There is a difference between the requirements for a hyperbolic pde like $$u_t + a u_x = 0$$ and for a purely parabolic pde like $$u_t = u_{xx}$$ Suppose the solutions are smooth and you approximate them by some finite difference method. Then in case of hyperbolic problem, the maximum error in the numerical solution depends on the time interval of computation $$[0,T]$$. The numerical error increases with time, so if you want to do long time simulation, it pays to use a high order method so the error at $$t=T$$ is still sufficiently small. The situation is similar for more general hyperbolic pde and even with non-smooth solutions though it is harder to construct uniformly high order schemes.

For purely parabolic problem like heat equation, if $$T$$ is large enough, the maximum error in the numerical solution is independent of $$T$$. This is because the solutions themselves decay, and the numerical error, though it grows initially, will also start to decay at large times. It is then enough to use second order accurate schemes for parabolic problems.

This can be seen by doing a Fourier analysis, see

Gustafsson, Kreiss, Oliger, "Time dependent problems and difference method", First Edition, Chapter 3.

• Thanks for giving such insights. So, I am indeed going to solve for long times. I am a little confused between choosing the schemes since one reference does use the Godunov method but haven't mentioned the accuracy. On the other hand, another paper recommend higher order schemes in order to get better accuracy. – Ronnie1993 May 7 at 15:35
• If you expect your solution to contain a discontinuity then formally you cannot reach anything higher than first order accuracy and that can only be reached by using Glimm’s method. This does not mean that something like WENO is not useful but you have to be careful. Even more problematic arguably is when you have discontinuities in coefficients where high-order accuracy can sometimes also not be advantageous. Instead refinement in space in these cases are more computationally advantageous. – Kyle Mandli May 11 at 13:03