[navtechdel]

I am trying to verify whether a scheme(for solving linear advection equation) is of third order or not. When I give the initial condition of a square pulse(1 in some part of domain and 0 elsewhere), I get order of roughly 1 and when i use Gaussian, I get an order of 4.

Does, the scheme's order depend on the initial condition provided?


To calculate the order, I am using L1 norm.

[mprinkey]

Most limited advection schemes necessarily drop to first order in the neighborhood of discontinuities. You will need to give us more details on your interpolation and limiting schemes for use to give you more definitive advice.

[FMDenaro]

Originally Posted by navtechdel

I am trying to verify whether a scheme(for solving linear advection equation) is of third order or not. When I give the initial condition of a square pulse(1 in some part of domain and 0 elsewhere), I get order of roughly 1 and when i use Gaussian, I get an order of 4.

Does, the scheme's order depend on the initial condition provided?


To calculate the order, I am using L1 norm.

It is theoretically wrong to do an accuracy order study using a non regular function. The local truncation error is expressed by derivatives so that you cannot get the correct term when they go to infinite.
Accuracy can be evaluated out from the singular point. Have a look to the book of Leveque where this issue is discussed.

[navtechdel]

This scheme has three level stencil. The update equation is given as

u(i,2t) = c1 * u(i,t) + c2 * u(i-1,t) + c3 * u(i,0) + c4 * u(i-1,0) + c5 * u(i-2,0)
where u(a,b) represents the value at grid point a at time b and c1,c2,... are some weights(fixed, depends on courant number).

Right now, I am not using any limiter. The paper says it is third order scheme, which is what I am trying to verify.

[mprinkey]

I'm assuming that you have a few typos in your equation and that you are using a five-point stencil finite difference method. Are the coefficients computed based on some nonlinear functions of the u() values? If so, that has the elements of a (C)WENO scheme or similar nonlinear differencing scheme. Maybe you can just give us the reference to the paper providing the scheme.

[FMDenaro]

Originally Posted by navtechdel

This scheme has three level stencil. The update equation is given as

u(i,2t) = c1 * u(i,t) + c2 * u(i-1,t) + c3 * u(i,0) + c4 * u(i-1,0) + c5 * u(i-2,0)
where u(a,b) represents the value at grid point a at time b and c1,c2,... are some weights(fixed, depends on courant number).

Right now, I am not using any limiter. The paper says it is third order scheme, which is what I am trying to verify.

Given such type of scheme (I haven't checked if it is correct or not), you can very easily determine analytically the accuracy order, just use the Taylor expansion for spatial and temporal nodes and determine this way the local truncation error.

[navtechdel]

The link to the paper:

https://www.google.co.in/url?sa=t&rc...FsHwLQ&cad=rja