Whether the three-dimensional incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This work attempts to provide an affirmative answer to this long-standing open question from a numerical point of view by presenting a class of potentially singular solutions to the Euler equations computed in axisymmetric geometries. The solutions satisfy a periodic boundary condition along the axial direction and a no-flow boundary condition on the solid wall. The equations are discretized in space using a hybrid 6th-order Galerkin and 6th-order finite difference method on specially designed adaptive (moving) meshes that are dynamically adjusted to the evolving solutions. With a maximum effective resolution of over $(3 × 10^{12})^2$ near the point of the singularity, we are able to advance the solution up to $\tau_2 = 0.003505$ and predict a singularity time of $t_s ≈ 0.0035056$, while achieving a pointwise relative error of $O(10^{−4})$ in the vorticity vector ω and observing a $(3 × 10^8)$-fold increase in the maximum vorticity $\|ω\|_{\infty}$. The numerical data are checked against all major blowup/non-blowup criteria, including Beale–Kato–Majda, Constantin–Fefferman–Majda, and Deng–Hou–Yu, to confirm the validity of the singularity. A local analysis near the point of the singularity also suggests the existence of a self-similar blowup in the meridian plane.

In this paper, we design a new class of central compact schemes based on the cell-centered compact schemes of Lele {[}S.K. Lele, Compact finite difference schemes with spectral-like resolution, J. Comput. Phys. 103 (1992) 16-42{]}. These schemes equate a weighted sum of the nodal derivatives of a smooth function to a weighted sum of the function on both the grid points (cell boundaries) and the cell-centers. In our approach, instead of using a compact interpolation to compute the values on cell-centers, the physical values on these half grid points are stored as independent variables and updated using the same scheme as the physical values on the grid points. This approach increases the memory requirement but not the computational costs. Through systematic Fourier analysis and numerical tests, we observe that the schemes have excellent properties of high order, high resolution and low dissipation. It is an ideal class of schemes for the simulation of multi-scale problems such as aeroacoustics and turbulence.

No abstract uploaded!

Finding the stationary states of a free energy functional is an important problem in phase field crystal (PFC) models. Many efforts have been devoted to designing numerical schemes with energy dissipation and mass conservation properties. However, most existing approaches are time-consuming due to the requirement of small effective step sizes. In this paper, we discretize the energy functional and propose efficient numerical algorithms for solving the constrained nonconvex minimization problem. A class of gradient-based approaches, which are the so-called adaptive accelerated Bregman proximal gradient (AA-BPG) methods, is proposed, and the convergence property is established without the global Lipschitz constant requirements. A practical Newton method is also designed to further accelerate the local convergence with convergence guarantee. One key feature of our algorithms is that the energy dissipation and mass conservation properties hold during the iteration process. Moreover, we develop a hybrid acceleration framework to accelerate the AA-BPG methods and most of the existing approaches through coupling with the practical Newton method. Extensive numerical experiments, including two three-dimensional periodic crystals in the Landau--Brazovskii (LB) model and a two-dimensional quasicrystal in the Lifshitz--Petrich (LP) model, demonstrate that our approaches have adaptive step sizes which lead to a significant acceleration over many existing methods when computing complex structures.

We develop an alternative formulation of conservative finite difference weighted essentially non-oscillatory (WENO) schemes to solve conservation laws. In this formulation, the WENO interpolation of the solution and its derivatives are used to directly construct the numerical flux, instead of the usual practice of reconstructing the flux functions. Even though this formulation is more expensive than the standard formulation, it does have several advantages. The first advantage is that arbitrary monotone fluxes can be used in this framework, while the traditional practice of reconstructing flux functions can be applied only to smooth flux splitting. The second advantage, which is fully explored in this paper, is that a narrower effective stencil is used compared with previous high order finite difference WENO schemes based on the reconstruction of flux functions, with a Lax-Wendroff time discretization. We will describe the scheme formulation and present numerical tests for one- and two-dimensional scalar and system conservation laws demonstrating the designed high order accuracy and non-oscillatory performance of the schemes constructed in this paper.