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We propose an efficient semi-Lagrangian Characteristic Mapping (CM) method for solving the three-dimensional (3D) incompressible Euler equations. This method evolves advected quantities by discretizing the flow map associated with the velocity field. Using the properties of the Lie group of volume preserving diffeomorphisms SDiff, long-time deformations are computed from a composition of short-time submaps which can be accurately evolved on coarse grids. This method is a fundamental extension to the CM method for two-dimensional incompressible Euler equations [51]. We take a geometric approach in the 3D case where the vorticity is not a scalar advected quantity, but can be computed as a differential 2-form through the pullback of the initial condition by the characteristic map. This formulation is based on the Kelvin circulation theorem and gives point-wise a Lagrangian description of the vorticity field. We demonstrate through numerical experiments the validity of the method and show that energy is not dissipated through artificial viscosity and small scales of the solution are preserved. We provide error estimates and numerical convergence tests showing that the method is globally third-order accurate.
We present a novel particle management method using the Characteristic Mapping framework. In the context of explicit evolution of parametrized curves and surfaces, the surface distribution of marker points created from sampling the parametric space i
The G-equation is a well-known model for studying front propagation in turbulent combustion. In this paper, we develop an efficient model reduction method for computing textcolor{black}{regular solutions} of viscous G-equations in incompressible stea
In this paper, an energy-based discontinuous Galerkin method for dynamic Euler-Bernoulli beam equations is developed. The resulting method is energy-dissipating or energy-conserving depending on the simple, mesh-independent choice of numerical fluxes
We deal with the Finite Element Tearing and Interconnecting Dual Primal (FETI-DP) preconditioner for elliptic problems discretized by the virtual element method (VEM). We extend the result of [22] to the three dimensional case. We prove polylogarithm
In this work we design a new domain decomposition method for the Euler equations in 2 dimensions. The basis is the equivalence via the Smith factorization with a third order scalar equation to whom we can apply an algorithm inspired from the Robin-Ro