In this paper, we present a conservative semi-Lagrangian finite-difference scheme for the BGK model. Classical semi-Lagrangian finite difference schemes, coupled with an L-stable treatment of the collision term, allow large time steps, for all the range of Knudsen number [17, 27, 30]. Unfortunately, however, such schemes are not conservative. Lack of conservation is analyzed in detail, and two main sources are identified as its cause. Firstly, when using classical continuous Maxwellian, conservation error is negligible only if velocity space is resolved with sufficiently large number of grid points. However, for a small number of grid points in velocity space such error is not negligible, because the parameters of the Maxwellian do not coincide with the discrete moments. Secondly, the non-linear reconstruction used to prevent oscillations destroys the translation invariance which is at the basis of the conservation properties of the scheme. As a consequence, the schemes show a wrong shock speed in the limit of small Knudsen number. To treat the first problem and ensure machine precision conservation of mass, momentum and energy with a relatively small number of velocity grid points, we replace the continuous Maxwellian with the discrete Maxwellian introduced in [22]. The second problem is treated by implementing a conservative correction procedure based on the flux difference form as in [26]. In this way we can construct conservative semi-Lagrangian schemes which are Asymptotic Preserving (AP) for the underlying Euler limit, as the Knudsen number vanishes. The effectiveness of the proposed scheme is demonstrated by extensive numerical tests.

We propose an adaptive threshold dynamics method for wetting problems in three space dimensions. The method is based on solving a linear heat equation and a thresholding step in each iteration. The heat equation is discretized by a cell-centered finite volume method on an adaptively refined mesh. An efficient technique for volume conservation is developed on the nonuniform meshes based on a quick-sorting operation. By this method, we compute some interesting wetting problems on complicated surfaces. Numerical results verify some recent theories for the apparent contact angle on rough and chemically patterned surfaces.

In this paper, we investigate the performance of the exponential time differencing (ETD) method applied to the rotating shallow water equations. Comparing with explicit time stepping of the same order accuracy in time, the ETD algorithms could reduce the computational time in many cases by allowing the use of large time step sizes while still maintaining numerical stability. To accelerate the ETD simulations, we propose a localized approach that synthesizes the ETD method and overlapping domain decomposition. By dividing the original problem into many subdomain problems of smaller sizes and solving them locally, the proposed approach could speed up the calculation of matrix exponential vector products. Several standard test cases for shallow water equations of one or multiple layers are considered. The results show great potential of the localized ETD method for high-performance computing because each subdomain problem can be naturally solved in parallel at every time step.

The simulation of Rayleigh waves is important in a variety of geophysical applications. The computational challenge is the fact that very fine mesh is necessary as the waves are concentrated at the free surface and decay exponentially away from the free surface. To overcome this challenge and to develop a robust high order scheme for the simulation of Rayleigh waves, we develop a mortar discontinuous Galerkin method with staggered hybridization. The use of the mortar technique allows one to use fine mesh in only a local region near the free surface, and use coarse mesh in most of the domain. This approach reduces the computational cost significantly. The staggered hybridization allows the preservation of the strong symmetry of the stress tensor without complicated construction of basis functions. In particular, the basis functions are piecewise polynomial without any continuity requirement, and the coupling of the basis functions is performed by using carefully chosen hybridized variables. The resulting scheme is explicit in time, and only local saddle point system is solved for each time step. We will present several benchmark problems to demonstrate the performance of the proposed method.

Global linear instability analysis is a powerful tool for the complex flow diagnosis. However, the methods used in the past would generally suffer from some disadvantages, either the excessive computational resources for the low-order methods or the tedious mathematical derivations for the high-order methods. The present work proposed a CFD-aided Galerkin methodology which combines the merits from both the low-order and high-order methods, where the expansion on proper basis functions is preserved to ensure a small matrix size, while the differentials, incompressibility constraints and boundary conditions are realized by applying the low-order linearized Navier-Stokes equation solvers on the basis functions on a fine grid. Several test cases have shown that the new method can get satisfactory results for one-dimensional, two-dimensional and three-dimensional flow problems and also for the problems with complex geometries and boundary conditions.

This paper is devoted to a multi-mesh-scale approach for describing the dynamic behaviors of thin geophysical mass flows on complex topographies. Because the topographic surfaces are generally non-trivially curved, we introduce an appropriate local coordinate system for describing the flow behaviors in an efficient way. The complex surfaces are supposed to be composed of a finite number of triangle elements. Due to the unequal orientation of the triangular elements, the distinct flux directions add to the complexity of solving the Riemann problems at the boundaries of the triangular elements. Hence, a vertex-centered cell system is introduced for computing the evolution of the physical quantities, where the cell boundaries lie within the triangles and the conventional Riemann solvers can be applied. Consequently, there are two mesh scales: the element scale for the local topographic mapping and the vertex-centered cell scale for the evolution of the physical quantities. The final scheme is completed by employing the HLL-approach for computing the numerical flux at the interfaces. Three numerical examples and one application to a large-scale landslide are conducted to examine the performance of the proposed approach as well as to illustrate its capability in describing the shallow flows on complex topographies.

In this study, the dynamics of elliptic membrane shell model has been proposed and discussed numerically for the first time. Firstly, we show that the solution of this model exists and is unique. Secondly, we consider spatial and time discretizations of the time-dependent elliptic membrane shell by finite element method and Newmark scheme, respectively. Then, the corresponding existence, uniqueness, stability, convergence and a priori error estimate are given. Finally, we present numerical results involving a portion of an ellipsoidal shell and a portion of a spherical shell to verify the efficiency and convergence of the numerical scheme.

This article is devoted to three quadrature methods for the rapid solution of stochastic time-dependent Maxwell's equations with uncertain permittivity, permeability and initial conditions. We develop the mathematical analysis of the error estimate for single level Monte Carlo method, multi-level Monte Carlo method, and the quasi-Monte Carlo method. The theoretical results are supplemented by numerical experiments.

In this paper, a class of high order numerical schemes is proposed to solve the nonlinear parabolic equations with variable coefficients. This method is based on our previous work [11] for convection-diffusion equations, which relies on a special kernel-based formulation of the solutions and successive convolution. However, disadvantages appear when we extend the previous method to our equations, such as inefficient choice of parameters and unprovable stability for high-dimensional problems. To overcome these difficulties, a new kernel-based formulation is designed to approach the spatial derivatives. It maintains the good properties of the original one, including the high order accuracy and unconditionally stable for one-dimensional problems, hence allowing much larger time step evolution compared with other explicit schemes. In addition, without extra computational cost, the proposed scheme can enlarge the available interval of the special parameter in the formulation, leading to less errors and higher efficiency. Moreover, theoretical investigations indicate that it is unconditionally stable for multi-dimensional problems as well. We present numerical tests for one- and two-dimensional scalar and system, demonstrating the designed high order accuracy and unconditionally stable property of the scheme.

In this paper, the discrete unified gas-kinetic scheme (DUGKS) is extended to the convection heat transfer in porous media at representative elementary volume (REV) scale, where the changes of velocity and temperature fields are described by two kinetic equations. The effects from the porous medium are incorporated into the method by including the porosity into the equilibrium distribution function, and adding a resistance force in the kinetic equation for the velocity field. The proposed method is systematically validated by several canonical cases, including the mixed convection in porous channel, the natural convection in porous cavity, and the natural convection in a cavity partially filled with porous media. The numerical results are in good agreement with the benchmark solutions and the available experimental data. It is also shown that the coupled DUGKS yields a second-order accuracy in both temporal and spatial spaces.

In this study, natural convection flow in a porous cavity with sinusoidal temperature distribution has been analyzed by a new double multi relaxation time (MRT) Lattice Boltzmann method (LBM). We consider a copper/water nanofluid filling a porous cavity. For simulating the temperature and flow fields, D2Q5 and D2Q9 lattices are utilized respectively, and the effects of different Darcy numbers (Da) (0.001-0.1) and various Rayleigh numbers (Ra) ($10^3$-$10^5$) for porosity ($ε$) between 0.4 and 0.9 have been considered. Phase deviation ($θ$) changed from 0 to $π$ and the volume fraction of nanoparticles (Ø) varied from 0 to 6%. The present results show a good agreement with the previous works, thus confirming the reliability the new numerical method proposed in this paper. It is indicated that the heat transfer rate increases at increasing Darcy number, porosity, Rayleigh number, the volume fraction of nanoparticles and phase deviation. However, the most sensitive parameter is the Rayleigh number. The maximum Nusselt deviation is 10%, 32% and 33% for Ra=$10^3$, $10^4$ and $10^5$, respectively, with $ε = 0.4$ to $ε = 0.9$. It can be concluded that the effect of Darcy number on the heat transfer rate increases at increasing Rayleigh number, yielding a maximum enhancement of the average Nusselt number around 12% and 61% for Ra=$10^3$ and Ra=$10^5$, respectively.