[sundials-users] Solving a large system of algebraic and differential equations with IDA and periodic boundary conditions

[David Halpern]

Hi,

I’m looking for some advice regarding performance optimization in solving a large system of coupled algebraic and time-dependent differential equations using IDA.

The system involves two unknown functions, h(x,t) and f(x,t). The time-dependent PDE governs h(x,t), which is coupled to f(x,t); however, f(x,t) does not contain a time derivative. The equations are discretized with fourth-order finite differences and periodic boundary conditions.

Currently, I’m using the dense linear solver within IDA for Newton’s method. This setup works well for moderate problem sizes—for example, around 1000 grid points (2000 unknowns)—and parallelization with OpenMP performs adequately. However, when I increase the grid size (e.g., to 2000 points or more), parallelization offers little benefit, and I’d like to scale up further (to about 10,000 unknowns) for better resolution.

Given that the Jacobian is sparse (but not banded), I’m wondering if there are more efficient options. I know that the Sherman–Morrison formula can sometimes be used to convert a nearly banded matrix with periodic boundary conditions into a truly banded one.

Would using a sparse solver such as KLU or SuperLU improve performance in this case? Or are there other strategies or solvers better suited for this type of problem?

Any suggestions or insights would be greatly appreciated.

Best regards,
David Halpern

David Halpern, PhD

Professor

Department of Mathematics

University of Alabama

Tuscaloosa AL 35487

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[Daniel Reynolds]

Hi David,

I recommend that you try the KLU linear solver for this case, since it should do an excellent job of retaining sparsity in its LU factors due to the nearly banded structure.  The main challenge for most users when switching from dense or banded to sparse is that SUNDIALS requires the user to construct the Jacobian themselves, instead of providing utility routines that approximate this directly.  From your description of the problem, it sounds as though the Jacobian might be straightforward to construct, and if you're already doing this for the dense solver then you may have little difficulty transitioning to sparse.

Daniel R. Reynolds (he/him)

Professor, Dept of Mathematics and Statistics

University of Maryland Baltimore County, MP 429

410.455.2429

https://drreynolds.github.io