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What ODE solvers are optimized for solving stiff systems with sparse Jacobian? Such systems appear, for instance, when a parabolic PDE is discretized in space using typical finite difference or finite element methods.

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    $\begingroup$ ODE solvers call user functions and sparsity is in the latter. So I do not think ODE solvers need to be aware of sparsity. $\endgroup$ – Hui Zhang Oct 21 '14 at 8:49
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    $\begingroup$ I think the place where sparsity (of the Jacobian) would matter is in the solution of algebraic equations for implicit solvers. So the question is really about what linear or nonlinear solvers are called from within the ODE solver. I'm pretty sure the time-stepping methods in PETSc would be designed with sparsity in mind. $\endgroup$ – David Ketcheson Oct 22 '14 at 5:23

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