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A new type of systematic approach to study the incompressible Euler equations numerically via the vanishing viscosity limit is proposed in this work. We show the new strategy is unconditionally stable that the $L^2$-energy dissipates and $H^s$-norm is uniformly bounded in time without any restriction on the time step. Moreover, first-order convergence of the proposed method is established including both low regularity and high regularity error estimates. The proposed method is extended to full discretization with a newly developed iterative Fourier spectral scheme. Another main contributions of this work is to propose a new integration by parts technique to lower the regularity requirement from $H^4$ to $H^3$ in order to perform the $L^2$-error estimate. To our best knowledge, this is one of the very first work to study incompressible Euler equations by designing stable numerical schemes via the inviscid limit with rigorous analysis. Furthermore, we will present both low and high regularity errors from numerical experiments and demonstrate the dynamics in several benchmark examples., Comment: 19 pages

Phase field models are gradient flows with their energy naturally dissipating in time. In order to preserve this property, many numerical schemes have been well-studied. In this paper we consider a well-known method, namely the exponential integrator method (EI). In the literature a few works studied several EI schemes for various phase field models and proved the energy dissipation by either requiring a strong Lipschitz condition on the nonlinear source term or certain $L^\infty$ bounds on the numerical solutions (maximum principle). However for phase field models such as the (non-local) Cahn-Hilliard equation, the maximum principle no longer exists. As a result, solving such models via EI schemes remains open for a long time. In this paper we aim to give a systematic approach on applying EI-type schemes to such models by solving the Cahn-Hilliard equation with a first order EI scheme and showing the energy dissipation. In fact second order EI schemes can be handled similarly and we leave the discussion in a subsequent paper. To our best knowledge, this is the first work to handle phase field models without assuming any strong Lipschitz condition or $L^\infty$ boundedness. Furthermore, we will analyze the $L^2$ error and present some numerical simulations to demonstrate the dynamics.

We introduce a new sinc-type molecular beam epitaxy model which is derived from a cosine-type energy functional. The landscape of the new functional is remarkably similar to the classical MBE model with double well potential but has the additional advantage that all its derivatives are uniformly bounded. We consider first order IMEX and second order BDF2 discretization schemes. For both cases we quantify explicit time step constraints for the energy dissipation which is in good accord with the practical numerical simulations. Furthermore we introduce a new theoretical framework and prove unconditional uniform energy boundedness with no size restrictions on the time step. This is the first unconditional (i.e. independent of the time step size) result for semi-implicit methods applied to the phase field models without introducing any artificial stabilization terms or fictitious variables., Comment: 19 pages

We consider a parabolic sine-Gordon model with periodic boundary conditions. We prove a fundamental maximum principle which gives a priori uniform control of the solution. In the one-dimensional case we classify all bounded steady states and exhibit some explicit solutions. For the numerical discretization we employ first order IMEX, and second order BDF2 discretization without any additional stabilization term. We rigorously prove the energy stability of the numerical schemes under nearly sharp and quite mild time step constraints. We demonstrate the striking similarity of the parabolic sine-Gordon model with the standard Allen-Cahn equations with double well potentials., Comment: 14 pages; to appear in "Numerical Mathematics: Theory, Methods and Applications"

The Oxygen Depletion problem is an implicit free boundary value problem. The dynamics allow topological changes in the free boundary. We show several mathematical formulations of this model from the literature and give a new formulation based on a gradient flow with constraint. All formulations are shown to be equivalent. We explore the possibilities for the numerical approximation of the problem that arise from the different formulations. We show a convergence result for an approximation based on the gradient flow with constraint formulation that applies to the general dynamics including topological changes. More general (vector, higher order) implicit free boundary value problems are discussed. Several open problems are described., Comment: 30 pages, 4 figures

Adaptive time stepping methods for metastable dynamics of the Allen Cahn and Cahn Hilliard equations are investigated in the spatially continuous, semi-discrete setting. We analyse the performance of a number of first and second order methods, formally predicting step sizes required to satisfy specified local truncation error $\sigma$ in the limit of small order parameter $\epsilon \rightarrow 0$ during meta-stable dynamics. The formal predictions are made under stability assumptions that include the preservation of the asymptotic structure of the diffuse interface, a concept we call profile fidelity. In this setting, definite statements about the relative behaviour of time stepping methods can be made. Some methods, including all so-called energy stable methods but also some fully implicit methods, require asymptotically more time steps than others.The formal analysis is confirmed in computational studies. We observe that some provably energy stable methods popular in the literature perform worse than some more standard schemes. We show further that when Backward Euler is applied to meta-stable Allen Cahn dynamics, the energy decay and profile fidelity properties for these discretizations are preserved for much larger time steps than previous analysis would suggest. The results are established asymptotically for general interfaces, with a rigorous proof for radial interfaces. It is shown analytically and computationally that for most reaction terms, Eyre type time stepping performs asymptotically worse due to loss of profile fidelity., Comment: 45 pages, four figures