[eozugurlu]

Dear All,
Is the Crank-Nicolson method for a parabolic pde with source term conditionally stable? Our source term is a typical rho*u*(1-u) where rho=0.023 (constant and small); our pde is u_t=a*u_xx+ rho*u*(1-u), u_x(x=0,t)=0, u_x(x=1,t)=0, u(0,t) is given as well.
When we try to show the stability of the Crank-Nicolson, first of all we take the linear form of the source term as rho*u.
We get the amplification factor as
g=(1-2*R sin^2(theta/2)+(1/2)*rho*Delta t)/(1+2*R sin^2(theta/2)-(1/2)*rho*Delta t)
where R=(a*Delta t)/(Delta x)^2.
To get |g|<=1 we get a conditional stability ???? Is this normal? Today, I have read an article "Stability analysis of Crank–Nicolson and Euler schemes
for time-dependent diffusion equations" by Cassio M. Oishi · Jin Y. Yuan ·
Jose A. Cuminato · David E. Stewart and they say that if the BC is explicit then the Crank-Nicolson is conditionally stable.

[FMDenaro]

Originally Posted by eozugurlu

Dear All,
Is the Crank-Nicolson method for a parabolic pde with source term conditionally stable? Our source term is a typical rho*u*(1-u) where rho=0.023 (constant and small); our pde is u_t=a*u_xx+ rho*u*(1-u), u_x(x=0,t)=0, u_x(x=1,t)=0, u(0,t) is given as well.
When we try to show the stability of the Crank-Nicolson, first of all we take the linear form of the source term as rho*u.
We get the amplification factor as
g=(1-2*R sin^2(theta/2)+(1/2)*rho*Delta t)/(1+2*R sin^2(theta/2)-(1/2)*rho*Delta t)
where R=(a*Delta t)/(Delta x)^2.
To get |g|<=1 we get a conditional stability ???? Is this normal? Today, I have read an article "Stability analysis of Crank–Nicolson and Euler schemes
for time-dependent diffusion equations" by Cassio M. Oishi · Jin Y. Yuan ·
Jose A. Cuminato · David E. Stewart and they say that if the BC is explicit then the Crank-Nicolson is conditionally stable.

Before addressing my opinion, I would be sure of your scheme for the source term. It is explicit or implicit?

[eozugurlu]

source term is treated as rho*(u_m^n+1 + u_m^n)/2 in Crank-Nicolson.

[FMDenaro]

Ok, assuming a convection-diffusion-production equation, you have to produce the stabilty regions like I computed here. If you disregard the convection you have one plane since G_max(alpha,rho) for any theta.

https://www.researchgate.net/publica...extFileContent

[eozugurlu]

you are not using Crank-Nicolson, your scheme is an explicit one.

[FMDenaro]

Originally Posted by eozugurlu

you are not using Crank-Nicolson, your scheme is an explicit one.

Yes and what is the issue? For implicit scheme You have only to divide the RHS by the expression the multiplies E_n+1 in the LHS and you have G exactly like for explicit scheme.

[FMDenaro]

This is what I found in your case

[eozugurlu]

Dear Dr. Denaro,

We get the amplification factor using the Crank-Nicolson method applied to
u_t=a*u_xx+ rho*u*(1-u)
as
g=(1-2*R sin^2(theta/2)+(1/2)*rho*Delta t)/(1+2*R sin^2(theta/2)-(1/2)*rho*Delta t)
where R=(a*Delta t)/(Delta x)^2.
Can you say that the Crank-Nicolson method is unconditionally stable?

[FMDenaro]

Originally Posted by eozugurlu

Dear Dr. Denaro,

We get the amplification factor using the Crank-Nicolson method applied to
u_t=a*u_xx+ rho*u*(1-u)
as
g=(1-2*R sin^2(theta/2)+(1/2)*rho*Delta t)/(1+2*R sin^2(theta/2)-(1/2)*rho*Delta t)
where R=(a*Delta t)/(Delta x)^2.
Can you say that the Crank-Nicolson method is unconditionally stable?

You cannot do that. The classical von Neumann stability analsys applies for the linear case, you cannot use the analsys as-it-is for the non linear case.

[eozugurlu]

I have stated that I applied it to the linear form in my first post

[eozugurlu]

Originally Posted by FMDenaro

This is what I found in your case

what can you say about the stability then? Is the ratio less than 1 for every Delta and Delta x?

[FMDenaro]

Originally Posted by eozugurlu

Dear Dr. Denaro,

We get the amplification factor using the Crank-Nicolson method applied to
u_t=a*u_xx+ rho*u*(1-u)
as
g=(1-2*R sin^2(theta/2)+(1/2)*rho*Delta t)/(1+2*R sin^2(theta/2)-(1/2)*rho*Delta t)
where R=(a*Delta t)/(Delta x)^2.
Can you say that the Crank-Nicolson method is unconditionally stable?

You wrote the non linear source term.

I wrote in my note the amplification factor for the linear source term, you can deduce the condition for G(alpha, beta) at the worst theta =k*h condition.