Optimization-based limiters for the spectral element method

We introduce a new family of optimization based limiters for the h–ph–p spectral element method. The native spectral element advection operator is oscillatory, but due to its mimetic properties it is locally conservative and has a monotone property with respect to element averages. We exploit this property to construct locally conservative quasimonotone and sign-preserving limiters. The quasimonotone limiter prevents all overshoots and undershoots at the element level, but is not strictly non-oscillatory. It also maintains quasimonotonicity even with the addition of a dissipation term such as viscosity or hyperviscosity. The limiters are based on a least-squares formulation with equality and inequality constraints and are local to each element. We evaluate the new limiters using a deformational flow test case for advection on the surface of the sphere. We focus on mesh refinement for moderate (p=3)(p=3) and high order (p=6)(p=6) elements. As expected, the spectral element method obtains its formal order of accuracy for smooth problems without limiters. For advection of fields with cusps and discontinuities, the high order convergence is lost, but in all cases, p=6p=6 outperforms p=3p=3 for the same degrees of freedom.