Unconditional stability of semi-implicit discretizations of singular flows
Provides theoretical stability guarantees for a widely used numerical method for singular flows, but reveals a fundamental limitation in accuracy that cannot be avoided.
The paper proves unconditional energy stability for semi-implicit discretizations of singular p-Laplace flows, but shows that error estimates depend critically on a regularization parameter, with numerical experiments confirming reduced convergence rates for smaller parameters.
A popular and efficient discretization of evolutions involving the singular $p$-Laplace operator is based on a factorization of the differential operator into a linear part which is treated implicitly and a regularized singular factor which is treated explicitly. It is shown that an unconditional energy stability property for this semi-implicit time stepping strategy holds. Related error estimates depend critically on a required regularization parameter. Numerical experiments reveal reduced experimental convergence rates for smaller regularization parameters and thereby confirm that this dependence cannot be avoided in general.