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Any source of FDM schemes for the transport equation starts by explaining that explicit central differences for the equation of the type $u_t+au_x=0$ can cause oscillations and unconditionally unstable. However, I am questioning about stability of two other schemes: Crank-Nicolson and Implicit with central differences for the space derivative. I will apply the VN stability analysis and calculate the amplification factor for both schemes. That is, I substitute $u_j^n$ with $e^{nk+ijh\omega}$. $k$ stands for the time step and $h$ for the space step. For the Crank Nicolson with central differences (if I did not make any mistake) I get the following amplification factor:

$$Q=\frac{1-i\frac{ak}{2h}\sin(h\omega)}{1+i\frac{ak}{2h}\sin(h\omega)}$$ And for the implicit in time central difference

$$Q=\frac{1}{1+i\frac{ak}{2h}\sin(h\omega)}$$

Next, I calculate the absolute value of both: start with CN:

$$|Q|=|\frac{(1-i\frac{ak}{2h}\sin(h\omega))(1-i\frac{ak}{2h}\sin(h\omega))}{(1+i\frac{ak}{2h}\sin(h\omega))(1-i\frac{ak}{2h}\sin(h\omega))}| $$ $$ =1+\left(\frac{ak}{2h}\sin(h\omega)\right)^2 $$ and for the implicit Euler: $$|Q|=|\frac{(1-i\frac{ak}{2h}\sin(h\omega))}{(1+i\frac{ak}{2h}\sin(h\omega))(1-i\frac{ak}{2h}\sin(h\omega))}|=|\frac{1-i\frac{ak}{2h}\sin(h\omega)}{1+\left(\frac{ak}{2h}\sin(h\omega)\right)^2}|=1 $$ From this math, Crank Nicolson has the amplification factor is greater that one and is not $\leq 1+Ck$ for some $C$, does it mean it has some possible instability problems and oscillations might occur? For the implicit, the amplification factor is one in magnitude which makes me think it might be unstable. Please clarify me if I am wrong but I would like to know if these schemes are prone to oscillations at all and when it might happen.

Kamil
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  • There are so many errors in this question. I started trying to fix them but it is almost hopeless. – David Ketcheson Jan 27 '13 at 17:47
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    I recalculated the amplification factor and I think this is what you mean, but it would be VERY helpful if you could point out where the mistakes are so I can work on them. Thanks. – Kamil Jan 27 '13 at 21:49
  • Kamil, I think the problem may be that you've got some fundamental misunderstandings of how stability works, as well as some bad mathematics in your equations. To pick the very first statement I take issue with: Well, I don't expect much from the explicit scheme and even in that case if the step is small enough the oscillations can be removed Unstable means unstable, in some situations there is no step small enough. You also can't "remove" oscillations without modifying the underlying equations that you are modeling. – Aron Ahmadia Jan 28 '13 at 00:39
  • @Aron: I did a number of error corrections. – Kamil Jan 28 '13 at 04:47
  • Kamil - @DavidKetcheson is a recognized expert in this topic, I hope his answer below is useful to you. I think in this case your question revealed some gaps in your fundamentals, making the problem a rather bad fit for our site's format. I'm going to close it now (this is not a bad thing), but I would suggest that you read Prof. Ketcheson's comments, review your materials, and ask again (perhaps a more specific question, with less forward assertions, but it was useful that you showed your work). – Aron Ahmadia Jan 28 '13 at 11:38

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This is not an answer. It is an attempt to point out some of the errors in the question.

Any source of FDM schemes for the transport equation starts by explaining that explicit central differences for the equation of the type $u_t+au_x=0$ can cause oscillations and unconditionally unstable.

The explicit centered difference scheme for this equation (in time and space) is often called Leapfrog, and it is one of the most widely used schemes for solving this and other wave equations. It is stable.

However, I am questioning about stability of two other schemes: Crank-Nicolson and Implicit with central differences for the space derivative.

"Implicit" is not a scheme. It is a (huge) set of schemes. Crank-Nicolson is an implicit scheme.

I will apply the VN stability analysis and calculate the amplification factor for both schemes. That is, I substitute $u_j^n$ with $e^{nk+ijh\omega}$. $k$ stands for the time step and $h$ for the space step. For the Crank Nicolson with central differences (if I did not make any mistake) I get the following amplification factor:

$$Q=\frac{1-i\frac{ak}{2h}\sin(h\omega)}{1+i\frac{ak}{2h}\sin(h\omega)}$$ And for the implicit in time central difference

$$Q=\frac{1}{1+i\frac{ak}{2h}\sin(h\omega)}$$

Next, I calculate the absolute value of both: start with CN:

$$|Q|=|\frac{(1-i\frac{ak}{2h}\sin(h\omega))(1-i\frac{ak}{2h}\sin(h\omega))}{(1+i\frac{ak}{2h}\sin(h\omega))(1-i\frac{ak}{2h}\sin(h\omega))}| $$ $$ =1+\left(\frac{ak}{2h}\sin(h\omega)\right)^2 $$ and for the implicit Euler: $$|Q|=|\frac{(1-i\frac{ak}{2h}\sin(h\omega))}{(1+i\frac{ak}{2h}\sin(h\omega))(1-i\frac{ak}{2h}\sin(h\omega))}|=|\frac{1-i\frac{ak}{2h}\sin(h\omega)}{1+\left(\frac{ak}{2h}\sin(h\omega)\right)^2}|=1 $$

Much of this is wrong. Therefore your conclusions below are wrong too.

From this math, Crank Nicolson has the amplification factor is greater that one and is not $\leq 1+Ck$ for some $C$, does it mean it has some possible instability problems and oscillations might occur? For the implicit, the amplification factor is one in magnitude which makes me think it might be unstable. Please clarify me if I am wrong but I would like to know if these schemes are prone to oscillations at all and when it might happen.

David Ketcheson
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