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March 27, 2010, 06:05 
How to implement implicit TVD limiter?

#1 
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kmlin
Join Date: Mar 2010
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Hi everyone,
As title, my question is how to implement the TVD scheme (Flux/Slope limiters) implicitly. For example, If we define F(n+1, j+1) as the function at n+1 time step and j+1 cell, we can represent the general oneparameter family of explicit and implicit schemes of the following form u(n+1, j) + lamda*theda*[ h(n+1,j+1/2)  h(n+1,j1/2)] = u(n,j)  lamda(1theda)[ h(n,j+1/2)  h(n,j1/2) ] where "lamda" is a constant, "theda" is used to control the scheme is explicit or implicit, and "h" is the flux function containing limiter such as MINMOD. My question is how to implement the MINMOD in h(n+1, j+1/2) in the implicit scheme. Should I just use the information of previous time step and evaluate it explicitly? Thanks. 

March 27, 2010, 11:31 
Jacobianfree method

#2 
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Jed Brown
Join Date: Mar 2009
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For a one dimensional scheme, it could be useful to assemble the matrix based on differentiating through the limiter, but in 2D and especially 3D, this will produce a Jacobian that's just too expensive to work with. Actually computing the stencil is somewhat messy to do by hand, so you may want to consider automatic or symbolic differentiation. I would recommend using a Jacobianfree method preconditioned by a firstorder operator. Use of differentiable limiters is useful in this context. A good paper on this:
Code:
@article{gropp2000globalized, title={{Globalized NewtonKrylovSchwarz algorithms and software for parallel implicit CFD}}, author={Gropp, W. and Keyes, D. and Mcinnes, L.C. and Tidriri, MD}, journal={International Journal of High Performance Computing Applications}, volume={14}, number={2}, pages={102}, year={2000} } 

March 27, 2010, 23:42 

#3  
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kmlin
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Quote:
If possible, would you please show a little bit more explicit on the formulation of MINMOD (or some other limiters)? I have totally no idea how to formulate a function with "max(0, min(1,r)" or (r+r) / (1+r) which is the flux limiter? I have found dozens of papers talking about limiters, but none of which shows how to do this formulation. Or where I can find some useful materials on this? Thanks Kmlin 

March 28, 2010, 08:52 

#4  
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Jed Brown
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