The stochastic Galerkin and stochastic collocation method are two state-of-the-art methods for solving partial differential equations (PDE) containing random coefficients. While the latter method, which is based on sampling, can straightforwardly be applied to nonlinear stochastic PDEs, this is nontrivial for the stochastic Galerkin method and approximations are required. In this paper, both methods are used for constructing high-order solutions of a nonlinear stochastic PDE representing the magnetic vector potential in a ferromagnetic rotating cylinder. This model can be used for designing solid-rotor induction machines in various machining tools. A precise design requires to take ferromagnetic saturation effects into account and uncertainty on the nonlinear magnetic material properties. Implementation issues of the stochastic Galerkin method are addressed and a numerical comparison of the computational cost and accuracy of both methods is performed. The stochastic Galerkin method requires in general less stochastic unknowns than the stochastic collocation approach to reach a certain level of accuracy, however at a higher computational cost.

In this paper, we introduce a novel hybrid variational model which generalizes the classical total variation method and the wavelet shrinkage method. An alternating minimization direction algorithm is then employed. We also prove that it converges strongly to the minimizer of the proposed hybrid model. Finally, some numerical examples illustrate clearly that the new model outperforms the standard total variation method and wavelet shrinkage method as it recovers better image details and avoids the Gibbs oscillations.

Conservative numerical methods are often used for simulations of fluid flows involving shocks and other jumps with the understanding that conservation guarantees reasonable treatment near discontinuities. This is true in that convergent conservative approximations converge to weak solutions and thus have the correct shock locations. However, correct shock location results from any discretization whose violation of conservation approaches zero as the mesh is refined. Here we investigate the case of the Euler equations for a single gas using the Jones-Wilkins-Lee (JWL) equation of state. We show that a quasi-conservative method can lead to physically realistic solutions which are devoid of spurious pressure oscillations. Furthermore, we demonstrate that under certain conditions, a quasi-conservative method can exhibit higher rates of convergence near shocks than a strictly conservative counterpart of the same formal order.

In this paper, we investigate initial boundary value problems of the space-time fractional diffusion equation and its numerical solutions. Two definitions, i.e., Riemann-Liouville definition and Caputo one, of the fractional derivative are considered in parallel. In both cases, we establish the well-posedness of the weak solution. Moveover, based on the proposed weak formulation, we construct an efficient spectral method for numerical approximations of the weak solution. The main contribution of this work are threefold: First, a theoretical framework for the variational solutions of the space-time fractional diffusion equation is developed. We find suitable functional spaces and norms in which the space-time fractional diffusion problem can be formulated into an elliptic weak problem, and the existence and uniqueness of the weak solution are then proved by using existing theory for elliptic problems. Secondly, we show that in the case of Riemann-Liouville definition, the well-posedness of the space-time fractional diffusion equation does not require any initial conditions. This contrasts with the case of Caputo definition, in which the initial condition has to be integrated into the weak formulation in order to establish the well-posedness. Finally, thanks to the weak formulation, we are able to construct an efficient numerical method for solving the space-time fractional diffusion problem.

It is well known that, as non-continuum gas flows through microscale porous media, the gas permeability derived from Darcy law is larger than the absolute permeability, which is caused by the so-called Klinkenberg effect or slippage effect. In this paper, an effective definition of Knudsen number for gas flows through square arrays of circular cylinders and a local boundary condition for non-continuum gas flows are first proposed, and then the multi-relaxation-time lattice Boltzmann equation including discrete effects on boundary condition is used to investigate Klinkenberg effect on gas flow through circular cylinders in square arrays. Numerical results show that the celebrated Klinkenberg equation is only correct for low Knudsen number, and second-order correction to Klinkenberg equation is necessary with the increase of Knudsen number. Finally, the present numerical results are also compared to some available results, and in general an agreement between them is observed.

An energy conserving discretization of the elastic wave equation in second order formulation is developed for a composite grid, consisting of a set of structured rectangular component grids with hanging nodes on the grid refinement interface. Previously developed summation-by-parts properties are generalized to devise a stable second order accurate coupling of the solution across mesh refinement interfaces. The discretization of singular source terms of point force and point moment tensor type are also studied. Based on enforcing discrete moment conditions that mimic properties of the Dirac distribution and its gradient, previous single grid formulas are generalized to work in the vicinity of grid refinement interfaces. These source discretization formulas are shown to give second order accuracy in the solution, with the error being essentially independent of the distance between the source and the grid refinement boundary. Several numerical examples are given to illustrate the properties of the proposed method.

We investigate problems of fluid structure interaction type and aim for a formulation that leads to a well posed problem and a stable numerical procedure. Our first objective is to investigate if the generally accepted formulations of the fluid structure interaction problem are the only possible ones. Our second objective is to derive a stable numerical coupling. To accomplish that we will use a weak coupling procedure and employ summation-by-parts operators and penalty terms. We compare the weak coupling with other common procedures. We also study the effect of high order accurate schemes. In multiple dimensions this is a formidable task and we start by investigating the simplest possible model problem available. As a flow model we use the linearized Euler equations in one dimension and as the structure model we consider a spring.

In this paper, we introduce a moment closure which is intended to provide a macroscopic approximation of the evolution of a particle distribution function, solution of a kinetic equation. This closure is of non-local type in the sense that it involves convolution or pseudo-differential operators. We show it is consistent with the diffusion limit and we propose numerical approximations to treat the non-local terms. We illustrate how this approach can be incorporated in complex models involving a coupling with hydrodynamic equations, by treating examples arising in radiative transfer. We pay a specific attention to the conservation of the total energy by the numerical scheme.

In the low-frequency fast multipole algorithm (LF-FMA) [19, 20], scalar addition theorem has been used to factorize the scalar Green's function. Instead of this traditional factorization of the scalar Green's function by using scalar addition theorem, we adopt the vector addition theorem for the factorization of the dyadic Green's function to realize memory savings. We are to validate this factorization and use it to develop a low-frequency vector fast multipole algorithm (LF-VFMA) for low-frequency problems. In the calculation of non-near neighbor interactions, the storage of translators in the method is larger than that in the LF-FMA with scalar addition theorem. Fortunately it is independent of the number of unknowns. Meanwhile, the storage of radiation and receiving patterns is linearly dependent on the number of unknowns. Therefore it is worthwhile for large scale problems to reduce the storage of this part. In this method, the storage of radiation and receiving patterns can be reduced by 25 percent compared with the LF-FMA.

The main goal of this paper is to develop the coupled double-distribution-function (DDF) lattice Boltzmann method (LBM) for simulation of subsonic and transonic turbulent flows. In the present study, we adopt the second-order implicit-explicit (IMEX) Runge-Kutta schemes for time discretization and the Non-Oscillatory and Non-Free-Parameters Dissipative (NND) finite difference scheme for space discretization. The Sutherland's law is used for expressing the viscosity of the fluid due to considerable temperature change. Also, the Spalart-Allmaras (SA) turbulence model is incorporated in order for the turbulent flow effect to be pronounced. Numerical experiments are performed on different turbulent compressible flows around a NACA0012 airfoil with body-fitted grid. Our numerical results are found to be in good agreement with experiment data and/or other numerical solutions, demonstrating the applicability of the method presented in this study to simulations of both subsonic and transonic turbulent flows.

We develop a numerical method for approximating the surface modes of sphere-like nanoparticles in the quasi-static limit, based on an expansion of (the angular part of) the potentials into spherical harmonics. Comparisons of the results obtained in this manner with exact solutions and with a perturbation ansatz prove that the scheme is accurate if the shape deviations from a sphere are not too large. The method allows fast calculations for large numbers of particles, and thus to obtain statistics for nanoparticles with random shape fluctuations. As an application we present some statistics for the distribution of resonances, polariziabilities, and dipole axes for particles with random perturbations.

In this paper, we use trigonometric polynomial reconstruction, instead of algebraic polynomial reconstruction, as building blocks for the weighted essentially non-oscillatory (WENO) finite difference schemes to solve hyperbolic conservation laws and highly oscillatory problems. The goal is to obtain robust and high order accurate solutions in smooth regions, and sharp and non-oscillatory shock transitions. Numerical results are provided to illustrate the behavior of the proposed schemes.

In this note, we propose a new method to cure numerical shock instability by hybriding different numerical fluxes in the two-dimensional Euler equations. The idea of this method is to combine a "full-wave" Riemann solver and a "less-wave" Riemann solver, which uses a special modified weight based on the difference in velocity vectors. It is also found that such blending does not need to be implemented in all equations of the Euler system. We point out that the proposed method is easily extended to other "full-wave" fluxes that suffer from shock instability. Some benchmark problems are presented to validate the proposed method.