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This paper deals with the implementation of high order semi-Lagrangian particle methods to handle high dimensional Vlasov–Poisson systems. It is based on recent developments in the numerical analysis of particle methods and the paper focuses on specific algorithmic features to handle large dimensions. The methods are tested with uniform particle distributions in particular against a recent multi-resolution wavelet based method on a 4D plasma instability case and a 6D gravitational case. Conservation properties, accuracy and computational costs are monitored. The excellent accuracy/cost trade-off shown by the method opens new perspective for accurate simulations of high dimensional kinetic equations by particle methods. [ABSTRACT FROM AUTHOR]

This paper presents a stable numerical algorithm for the Brinkman equations by using weak Galerkin (WG) finite element methods. The Brinkman equations can be viewed mathematically as a combination of the Stokes and Darcy equations which model fluid flow in a multi-physics environment, such as flow in complex porous media with a permeability coefficient highly varying in the simulation domain. In such applications, the flow is dominated by Darcy in some regions and by Stokes in others. It is well known that the usual Stokes stable elements do not work well for Darcy flow and vice versa. The challenge of this study is on the design of numerical schemes which are stable for both the Stokes and the Darcy equations. This paper shows that the WG finite element method is capable of meeting this challenge by providing a numerical scheme that is stable and accurate for both Darcy and the Stokes dominated flows. Error estimates of optimal order are established for the corresponding WG finite element solutions. The paper also presents some numerical experiments that demonstrate the robustness, reliability, flexibility and accuracy of the WG method for the Brinkman equations. [ABSTRACT FROM AUTHOR]

In this paper, a segregated algorithm is proposed to suppress high-frequency oscillations in the velocity field for incompressible flows. In this context, a new velocity formula based on a reconstruction of face fluxes is defined eliminating high-frequency errors. In analogy to the Rhie–Chow interpolation, this approach is equivalent to including a flux-based pressure gradient with a velocity diffusion in the momentum equation. In order to guarantee second-order accuracy of the numerical solver, a set of conditions are defined for the reconstruction operator. To arrive at the final formulation, an outlook over the state of the art regarding velocity reconstruction procedures is presented comparing them through an error analysis. A new operator is then obtained by means of a flux difference minimization satisfying the required spatial accuracy. The accuracy of the new algorithm is analyzed by performing mesh convergence studies for unsteady Navier–Stokes problems with analytical solutions. The stabilization properties of the solver are then tested in a problem where spurious numerical oscillations arise for the velocity field. The results show a remarkable performance of the proposed technique eliminating high-frequency errors without losing accuracy. [ABSTRACT FROM AUTHOR]

In this paper, some numerical algorithms (spectral collocation method, block spectral collocation method, boundary value method, block boundary value method, implicit Runge–Kutta method, diagonally implicit Runge–Kutta method and total variation diminishing Runge–Kutta method) are used to solve the highly oscillatory second-order initial value problems. We first derive these methods for the first-order initial value problems, and then extend these methods to the highly oscillatory nonlinear systems by matrix analysis methods. These new methods preserve the accuracy of the original methods and the main advantages of these new methods are low storage requirements and high efficiency. Extensive numerical results are presented to demonstrate the convergence properties of these methods. [ABSTRACT FROM AUTHOR]

Fourier extensions have been shown to be an effective means for the approximation of smooth, nonperiodic functions on bounded intervals given their values on an equispaced, or in general, scattered grid. Related to this method are two parameters. These are the extension parameter T (the ratio of the size of the extended domain to the physical domain) and the oversampling ratio η (the number of sampling nodes per Fourier mode). The purpose of this paper is to investigate how the choice of these parameters affects the accuracy and stability of the approximation. Our main contribution is to document the following interesting phenomenon: namely, if the desired condition number of the algorithm is fixed in advance, then the particular choice of such parameters makes little difference to the algorithm's accuracy. As a result, one is free to choose T without concern that it is suboptimal. In particular, one may use the value T=2 - which corresponds to the case where the extended domain is precisely twice the size of the physical domain - for which there is known to be a fast algorithm for computing the approximation. In addition, we determine the resolution power (points-per-wavelength) of the approximation to be equal to Tη, and address the trade-off between resolution power and stability. [ABSTRACT FROM AUTHOR]

Abstract: This is a research expository paper regarding superparameterization, a class of multi-scale numerical methods designed to cope with the intermittent multi-scale effects of inhomogeneous geophysical turbulence where energy often inverse-cascades from the unresolved scales to the large scales through the effects of waves, jets, vortices, and latent heat release from moist processes. Original as well as sparse space–time superparameterization algorithms are discussed for the important case of moist atmospheric convection including the role of multi-scale asymptotic methods in providing self-consistent constraints on superparameterization algorithms and related deterministic and stochastic multi-cloud parameterizations. Test models for the statistical numerical analysis of superparameterization algorithms are discussed both to elucidate the performance of the basic algorithms and to test their potential role in efficient multi-scale data assimilation. The very recent development of grid-free seamless stochastic superparameterization methods for geophysical turbulence appropriate for “eddy-permitting” mesoscale ocean turbulence is presented here including a general formulation and illustrative applications to two-layer quasigeostrophic turbulence, and another difficult test case involving one-dimensional models of dispersive wave turbulence. This last test case has randomly generated solitons as coherent structures which collapse and radiate wave energy back to the larger scales, resulting in strong direct and inverse turbulent energy cascades. [Copyright &y& Elsevier]

Abstract: Stable partitioned algorithms for fluid–structure interaction (FSI) problems are developed and analyzed in this two-part paper. Part I describes an algorithm for incompressible flow coupled with compressible elastic solids, while Part II discusses an algorithm for incompressible flow coupled with structural shells. The numerical approach described here for structural shells uses Robin (mixed) interface conditions for the pressure and velocity in the fluid which are derived directly from the governing equations. The resulting added-mass partitioned (AMP) algorithm is stable even for very light structures, requires no sub-iterations per time step, and is second-order accurate. The stability and accuracy of the AMP algorithm is evaluated for linearized FSI model problems. A normal mode analysis is performed to show that the new AMP algorithm is stable, even for the case of very light structures when added-mass effects are large. Exact traveling wave solutions are derived for the FSI model problems, and these solutions are used to verify the stability and accuracy of the corresponding numerical results obtained from the AMP algorithm for the cases of light, medium and heavy structures. A summary comparison of the AMP algorithm developed here and the one in Part I is provided. [Copyright &y& Elsevier]

• A linearly operator splitting algorithm is proposed for the nonlocal VCH equation. • The energy stabilities for both subproblems are proved. • The stability and convergence of the operator splitting algorithm are studied. • A semi-implicit SDC method is further used to improve time accuracy. Recently, the viscous Cahn-Hilliard (VCH) equation has been proposed as a phenomenological continuum model for phase separation in glass and polymer systems where intermolecular friction forces become important. Compared with the classical local VCH model, the nonlocal VCH model equipped with nonlocal diffusion operator can describe more practical phenomena for modeling phase transitions of microstructures in materials. This paper presents a high order fast explicit method based on operator splitting and spectral deferred correction (SDC) for solving the nonlocal VCH equation. We start with a second-order operator splitting spectral scheme, which is based on the Fourier spectral method and the strong stability preserving Runge-Kutta (SSP-RK) method. The scheme takes advantage of applying the fast Fourier transform (FFT) and avoiding nonlinear iteration. The stability and convergence analysis of the obtained numerical scheme are analyzed. To improve the temporal accuracy, the semi-implicit SDC method is then introduced. Various numerical simulations are performed to validate the accuracy and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]