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Register a new userA diffusion-driven Characteristic Mapping method for particle management
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Added by
Xi-Yuan Yin
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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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 is controlled by the area element of the parametrization function. As the surface evolves, the area element becomes uneven and the sampling, suboptimal. In this method we maintain the quality of the sampling by pre-composition of the parametrization with a deformation map of the parametric space. This deformation is generated by the velocity field associated to the diffusion process on the space of probability distributions and induces a uniform redistribution of the marker points. We also exploit the semigroup property of the heat equation to generate a submap decomposition of the deformation map which provides an efficient way of maintaining evenly distributed marker points on curves and surfaces undergoing extensive deformations.
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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 Petrov-Gelerkin (PG) method for a class of nonlocal convection-dominated diffusion problems. There are two main ingredients in our approach. First, we define the norm on the test space as induced by the trial space norm, i.e., the optimal test norm, so that the inf-sup condition can be satisfied uniformly independent of the problem. We show the well-posedness of a class of nonlocal convection-dominated diffusion problems under the optimal test norm with general assumptions on the nonlocal diffusion and convection kernels. Second, following the framework of Cohen et al.~(2012), we embed the original nonlocal convection-dominated diffusion problem into a larger mixed problem so as to choose an enriched test space as a stabilization of the numerical algorithm. In the numerical experiments, we use an approximate optimal test norm which can be efficiently implemented in 1d, and study its performance against the energy norm on the test space. We conduct convergence studies for the nonlocal problem using uniform $h$- and $p$-refinements, and adaptive $h$-refinements on both smooth manufactured solutions and solutions with sharp gradient in a transition layer. In addition, we confirm that the PG method is asymptotically compatible.
There are plenty of applications and analysis for time-independent elliptic partial differential equations in the literature hinting at the benefits of overtesting by using more collocation conditions than the number of basis functions. Overtesting not only reduces the problem size, but is also known to be necessary for stability and convergence of widely used unsymmetric Kansa-type strong-form collocation methods. We consider kernel-based meshfree methods, which is a method of lines with collocation and overtesting spatially, for solving parabolic partial differential equations on surfaces without parametrization. In this paper, we extend the time-independent convergence theories for overtesting techniques to the parabolic equations on smooth and closed surfaces.
A novel sharp interface ghost-cell based immersed boundary method has been proposed and its parameters have been optimized against an analytical model in diffusion applications. The proposed embedded constrained moving least-squares (ECMLS) algorithm minimizes the error of the interpolated concentration at the image point of the ghost point by applying a moving least-squares method on all internal nodes, near the ghost image point, and the associated mirrored image points of these internal nodes through the corresponding boundary conditions. Using an analytical model as a reference, the ECMLS algorithm is compared to the constrained moving least-squares (CMLS) algorithm and the staircase model using various grid sizes, interpolation basis functions, weight functions, and the penalty parameter of the constraint. It is found that using ECMLS algorithm in the investigated diffusion application, the incomplete quartic basis function yields the best performance while the quadratic, cubic, and bicubic basis functions also produce results better than the staircase model. It is also found that the linear and bilinear basis functions cannot produce results better than the staircase model in diffusion applications. It is shown that the optimal radius of the region of internal nodes used for interpolation scales with the logarithm of the boundary radius of curvature. It is shown that for the diffusion application, the proposed ECMLS algorithm produces lower errors at the boundary with better numerical stability over a wider range of basis functions, weight functions, boundary radius of curvature, and the penalty parameter than the CMLS algorithm.
We consider two parallel-in-time approaches applied to a (reaction) diffusion problem, possibly non-linear. In particular, we consider PFASST (Parallel Full Approximation Scheme in Space and Time) and space-time multilevel strategies. For both approaches, we start from an integral formulation of the continuous time-dependent problem. Then, a collocation form for PFASST and a discontinuous Galerkin discretization in time for the space-time multigrid are employed, resulting in the same discrete solution at the time nodes. Strong and weak scaling of both multilevel strategies is compared for varying order of the temporal discretization. Moreover, we investigate the respective convergence behavior for non-linear problems and highlight quantitative differences.
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