Search Results for « High-fidelity numerical methods »

Lack of conservation has been the biggest drawback in meshfree generalized finite difference methods (GFDMs). In this paper, we present a novel modification of classical meshfree GFDMs to include local balances which produce an approximate conservation of numerical fluxes. This numerical flux conservation is done within the usual moving least squares framework. Unlike Finite Volume Methods, it is based on locally defined control cells, rather than a globally defined mesh. We present the application of this method to an advection diffusion equation and the incompressible Navier - Stokes equations. Our simulations show that the introduction of flux conservation significantly reduces the errors in conservation in meshfree GFDMs.

Energy-conserving discretizations are widely regarded as a fundamental requirement for high-fidelity simulations of turbulent flows. A well-known approach to obtain semi-discrete conservation of energy in the inviscid limit is to employ the skew-symmetric splitting of the non-linear term. However, this approach has the drawback of being roughly twice as expensive as the computation of the divergence or advective forms alone. A novel time-advancement strategy that retains the conservation properties of skew-symmetric-based schemes at a reduced computational cost has been developed. This method is based on properly constructed Runge-Kutta schemes in which a different form (advective or divergence) for the convective term is adopted at each stage. A general theoretical framework has been developed to derive new schemes with prescribed accuracy on both solution and energy conservation. The technique has been first developed and fine-tuned on the Burgers' equation, and then applied to the incompressible Navier-Stokes equations. Simulations of homogeneous isotropic turbulence performed at the Center for Turbulence Research (CTR) in Stanford have demonstrated that the novel procedure provides the same robustness of the skew-symmetric form while halving the computational cost for the non-linear term.

We couple the L1 discretization for Caputo derivative in time with spectral Galerkin method in space to devise a scheme that solves quasilinear subdiffusion equations. Both the diffusivity and the source are allowed to be nonlinear functions of the solution. We prove method's stability and convergence with spectral accuracy in space. The temporal order depends on solution's regularity in time. Further, we support our results with numerical simulations that utilize parallelism for spatial discretization. Moreover, as a side result we find asymptotic exact values of error constants along with their remainders for discretizations of Caputo derivative and fractional integrals. These constants are the smallest possible which improves the previously established results from the literature.

Abstract A wide range of flows of practical interest occur in cylindrical geometries. In order to simulate such flows, an available compact finite‐difference simulation code [1] was adapted by introducing a mapping that expresses cylindrical coordinates as generalized coordinates. This formulation is conservative and avoids problems associated with the classical formulation of the Navier‐Stokes equations in cylindrical coordinates. The coordinate singularity treatment follows [2] and is modified for generalized coordinates. To retain high‐order numerical accuracy, a Fourier spectral method is employed in the azimuthal direction combined with mode clipping to alleviate time‐step restrictions due to a very fine grid spacing near the singularity at the axis (r = 0). An implementation of this scheme was successfully validated by a simulation of a tripolar vortex formation and by comparison with linear stability theory. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Left-invariant PDE-evolutions on the roto-translation group $SE(2)$ (and their resolvent equations) have been widely studied in the fields of cortical modeling and image analysis. They include hypo-elliptic diffusion (for contour enhancement) proposed by Citti & Sarti, and Petitot, and they include the direction process (for contour completion) proposed by Mumford. This paper presents a thorough study and comparison of the many numerical approaches, which, remarkably, is missing in the literature. Existing numerical approaches can be classified into 3 categories: Finite difference methods, Fourier based methods (equivalent to $SE(2)$-Fourier methods), and stochastic methods (Monte Carlo simulations). There are also 3 types of exact solutions to the PDE-evolutions that were derived explicitly (in the spatial Fourier domain) in previous works by Duits and van Almsick in 2005. Here we provide an overview of these 3 types of exact solutions and explain how they relate to each of the 3 numerical approaches. We compute relative errors of all numerical approaches to the exact solutions, and the Fourier based methods show us the best performance with smallest relative errors. We also provide an improvement of Mathematica algorithms for evaluating Mathieu-functions, crucial in implementations of the exact solutions. Furthermore, we include an asymptotical analysis of the singularities within the kernels and we propose a probabilistic extension of underlying stochastic processes that overcomes the singular behavior in the origin of time-integrated kernels. Finally, we show retinal imaging applications of combining left-invariant PDE-evolutions with invertible orientation scores.

The unsteady aerodynamic and aeroelastic behavior of a 2D wing section with and without flap is analyzed with Theodorsen theory and Unsteady Vortex Lattice Method (low fidelity), Euler (medium fidelity) and Reynolds-Averaged Navier Stokes (high fidelity) methods. The aeroelastic studies are carried out for linear cases and non-linear structural configurations presenting cubic stiffness and freeplay. The critical flutter speeds as well as the limit cycle oscillations present in the non-linear cases are compared. The methods show good agreement for the cases studied.

The aim of this work is to study the numerical solution of the nonlinear Schrodinger problem using a combination between Witt basis and finite difference approximations. We construct a discrete fundamental solution for the non-stationary Schrodinger operator and we show the convergence of the numerical scheme. Numerical examples are given at the end of the paper.