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We present a black-box method to numerically investigate the linear stabilityof arbitrary multi-physics problems. While the user just has to enter thesystem's residual in weak formulation, i.e. by a finite element method, allrequired discretized matrices are automatically assembled based on just-in-timegenerated and compiled highly performant C code. Based on this method, entirephase diagrams in the parameter space can be obtained by bifurcation trackingand continuation within minutes. Particular focus is put on problems withmoving domains, e.g. free surface problems in fluid dynamics, since a movingmesh introduces a plethora of complicated nonlinearities to the system. Bysymbolic differentiation before the code generation, however, these moving meshproblems are made accessible to bifurcation tracking methods. In a second step,our method is generalized to investigate symmetry-breaking instabilities ofaxisymmetric stationary solutions by effectively utilizing the symmetry of thebase state. Each bifurcation type is validated on the basis of results reportedin the literature on versatile fluid dynamics problems.

Bifurcation tracking on moving meshes and with consideration of azimuthal symmetry breaking instabilities