Preconditioning and Linearly Implicit Time Integration for the Serre-Green-Naghdi Equations

The treatment of the differential PDE constraint poses a key challenge in computing the numerical solution of the Serre-Green-Naghdi (SGN) equations. In this work, we introduce a constant coefficient preconditioner for the SGN constraint operator and prove rigorous bounds on the preconditioned conditioning number. The conditioning bounds incorporate the effects of bathymetry in two dimensions, are quasi-optimal within a class of constant coefficient operators, highlight fundamental scalings for a loss of conditioning, and ensure mesh independent performance for iterative Krylov methods. Utilizing the conditioning bounds, we devise and test two time integration strategies for solving the full SGN equations. The first class combines classical explicit time integration schemes (4th order Runge-Kutta and 2nd--4th order Adams-Bashforth) with the new preconditioner. The second is a linearly implicit scheme where the differential constraint is split into a constant coefficient implicit part and remaining (stiff) explicit part. The linearly implicit methods require a single linear solve of a constant coefficient operator at each time step. We provide a host of computational experiments that validate the robustness of the preconditioners, as well as full solutions of the SGN equations including solitary waves traveling over an underwater shelf (in 1d) and a circular bump (in 2d).