In this paper, we extend the method (Fu et al., [1]) to anisotropic meshes by introducing an adaptive SPH (ASPH) concept with ellipsoidal kernels. First, anisotropic target feature-size and density functions, taking into account the effects of singularities, are defined based on the level-set methodology. Second, ASPH is developed such that the particle distribution relaxes towards the target functions. In order to prevent SPH particles from escaping the mesh generation regions, a ghost surface particle method is proposed in combination with a tailored interaction strategy. Necessary adaptations of supporting numerical algorithms, such as fast neighbor search, for enforcing mesh anisotropy are addressed. Finally, unstructured meshes are generated by an anisotropic Delaunay triangulation conforming to the Riemannian metrics for the resulting particle configuration. The performance of the proposed method is demonstrated by a set of benchmark cases.

Dealing with strong shocks while retaining low numerical dissipation traditionally has been one of the major challenges for high order methods like discontinuous Galerkin (DG). In the literature, shock capturing models have been designed for DG based on various approaches, such as slope limiting, (H)WENO reconstruction, a posteriori sub-cell limiting, and artificial viscosity, among which a subclass of artificial viscosity methods are compared in the present work. Four models are evaluated, including a dilation-based model, a highest modal decay model, an averaged modal decay model, and an entropy viscosity model. Performance for smooth, non-smooth and broadband problems are examined with typical one- and two-dimensional cases.

We study the mechanism of the gauge cooling technique to stabilize the complex Langevin method in the one-dimensional periodic setting. In this case, we find the exact solutions for the gauge transform which minimizes the Frobenius norm of link variables. Thereby, we derive the underlying stochastic differential equations by continuing the numerical method with gauge cooling, and thus provide a number of insights on the effects of gauge cooling. A specific case study is carried out for the Polyakov loop model in SU(2) theory, in which we show that the gauge cooling may help form a localized distribution to guarantee there is no excursion too far away from the real axis. 

We propose a decoupled and positivity-preserving discrete duality finite volume (DDFV) scheme for anisotropic diffusion problems on polyhedral meshes with star-shaped cells and planar faces. Under the generalized DDFV framework, two sets of finite volume (FV) equations are respectively constructed on the dual and primary meshes, where the ones on the dual mesh are derived from the ingenious combination of a geometric relationship with the construction of the cell matrix. The resulting system on the dual mesh is symmetric and positive definite, while the one on the primary mesh possesses an M-matrix structure. To guarantee the positivity of the two categories of unknowns, a cutoff technique is introduced. As for the local conservation, it is conditionally maintained on the dual mesh while strictly preserved on the primary mesh. More interesting is that the FV equations on the dual mesh can be solved independently, so that the two sets of FV equations are decoupled. As a result, no nonlinear iteration is required for linear problems and a general nonlinear solver could be used for nonlinear problems. In addition, we analyze the well-posedness of numerical solutions for linear problems. The properties of the presented scheme are examined by numerical experiments. The efficiency of the Newton method is also demonstrated by comparison with those of the fixed-point iteration method and its Anderson acceleration.

We present an efficient and conservative Eulerian-Lagrangian method for solving two-dimensional hydrostatic multilayer shallow water flows with mass exchange between the vertical layers. The method consists of a projection finite volume method for the Eulerian stage and a method of characteristics to approximate the numerical fluxes for the Lagrangian stage. The proposed method is simple to implement, satisfies the conservation property and it can be used for multilayer shallow water equations on non-flat bathymetry including eddy viscosity and Coriolis forces. It offers a novel method of calculating stratified vertical velocities without the use of the Navier-Stokes equations. Numerical results are presented for several examples and the obtained results for a free-surface flow problem are in close agreement with the analytical solutions. We also test the performance of the proposed method for a test example of wind-driven flows with recirculation

In this paper, we study the multiscale computations for the Maxwell– Schrödinger system with rapidly oscillating coefficients under the dipole approximation that describes light-matter interaction in heterogeneous nanostructures. The multiscale asymptotic method and an associated numerical algorithm for the system are presented. We propose an alternating Crank–Nicolson finite element method for solving the homogenized Maxwell–Schödinger system and prove the existence of solutions to the discrete system. Numerical examples are given to validate the efficiency and accuracy of the algorithm.

In this paper, we present a new two-stage fourth-order finite difference weighted compact nonlinear scheme (WCNS) for hyperbolic conservation laws with special application to compressible Euler equations. To construct this algorithm, apart from the traditional WCNS for the spatial derivative, it was necessary to first construct a linear compact/explicit scheme utilizing time derivative of flux at midpoints, which, in turn, was solved by a generalized Riemann solver. Combining these two schemes, the fourth-order time accuracy was achieved using only the two-stage time-stepping technique. The final algorithm was numerically tested for various one-dimensional and two-dimensional cases. The results demonstrated that the proposed algorithm had an essentially similar performance as that based on the fourth-order Runge-Kutta method, while it required 25 percent less computational cost for one-dimensional cases, which is expected to decline further for multidimensional cases.

High-energy X-ray radiography is a measuring technique for quantitative measurement and diagnosis of the object and its internal structure. Tomographic reconstruction determines the geometric and physical properties of the object according to the energy distribution on the imaging plane. Considering the noise and blur in the process of radiographing, we construct a general reconstruction model for the axisymmetric single image photographic system. This inverse problem is then cast within a statistical framework in order to compute volumetric object densities from X-ray radiographs and to quantify uncertainties in the reconstruction. A hierarchical Bayesian model is developed with a likelihood based on a Gaussian noise model and with priors placed on the unknown nonnegative density profile, the precision matrix, and two scale parameters. This results in a joint posterior distribution, which can be readily sampled using the Markov chain Monte Carlo (MCMC) method. To study the role of hyperparameters and their sensitivity analysis, a wide variety of tests were conducted which led to a number of definitive conclusions. Results of the density reconstructions and pointwise uncertainty estimates are presented for simulated signals with various physical factors in the imaging process included.

The Poisson-Nernst-Planck (PNP) system is a widely accepted model for simulation of ionic channels. In this paper, we design, analyze, and numerically validate a second order unconditional positivity-preserving scheme for solving a reduced PNP system, which can well approximate the three dimensional ion channel problem. Positivity of numerical solutions is proven to hold true independent of the size of time steps and the choice of the Poisson solver. The scheme is easy to implement without resorting to any iteration method. Several numerical examples further confirm the positivity-preserving property, and demonstrate the accuracy, efficiency, and robustness of the proposed scheme, as well as the fast approach to steady states.

Radial basis function (RBF) collocation methods (RBFCMs) are applied to fourth order boundary value problems (BVPs). In particular, we consider the classical Kansa method and the method of approximate particular solutions (MAPS). In the proposed approach we include some so-called ghost points which are located inside and outside the domain of the problem. The inclusion of these points is shown to improve the accuracy and the stability of the collocation methods. An appropriate value of the shape parameter in the RBFs used is obtained using either the leave-one-out cross validation (LOOCV) algorithm or Franke's formula. We present and analyze the results of several numerical tests.

A two-dimensional front-tracking method is developed for handling complex shape interfaces satisfying the volume conservation. In order to validate the proposed front-tracking method, a complete convergence study is carried out on several analytical test cases for which the interface is widely stretched and deformed. Comparisons to different existing approaches show that our front-tracking method is second order accurate in space with lower errors than existing interface tracking techniques of the literature.
We also propose an original marker advection method which takes into account the jump relations valid at interface in order to deal with the contrast of physical properties encountered in two-phase flow simulations. The conservative front-tracking method computed in this work is shown to be able to describe interfaces with high accuracy even for poorly resolved Eulerian grids.