Large-Scale Continuation and Numerical Bifurcation for Partial Differential Equations

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Abstract
In this paper the problem of computing bifurcation diagrams for large-scale nonlinear parameter-dependent steady state systems which arise following the spatial discretization of semilinear PDEs is investigated. A continuation algorithm which employs a preconditioned version of the recursive projection method (RPM) is presented. The RPM is often expensive when it is used in conjunction with the numerical method of lines. Preconditioning the Jacobian of the underlying fixed point operator results in an algorithm (the preconditioned recursive projection method (PRPM)) which is capable of efficiently computing equilibrium solution diagrams of large stiff systems. For many PDE problems the PRPM is a fast and effective means of detecting both steady state and Hopf bifurcation along a branch of solutions. A description of the performance of the PRPM when applied to two numerical examples is given.

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