Equilibrium or stationary solutions usually proceed through the exact balance between hyperbolic transport terms and source terms. Such equilibrium solutions are affected by truncation errors that prevent any classical numerical scheme from capturing the evolution of small amplitude waves of physical significance. In order to overcome this problem, we compare two commonly adopted strategies: going to very high order and reduce drastically the truncation errors on the equilibrium solution, or design a specific scheme that preserves by construction the equilibrium exactly, the so-called well-balanced approach. We present a modern numerical implementation of these two strategies and compare them in details, using hydrostatic but also dynamical equilibrium solutions of several simple test cases. Finally, we apply our methodology to the simulation of a protoplanetary disc in centrifugal equilibrium around its star and model its interaction with an embedded planet, illustrating in a realistic application the strength of both methods.

We study the stability and dynamic transitions of the western boundary currents in a rectangular closed basin. By reducing the infinite dynamical system to a finite dimensional one via center manifold reduction, we derive a non-dimensional transition number that determines the types of dynamical transition. We show by careful numerical evaluation of the transition number that both continuous transitions (supercritical Hopf bifurcation) and catastrophic transitions (subcritical Hopf bifurcation) can happen at the critical Reynolds number, depending on the aspect ratio and stratification. The regions separating the continuous and catastrophic transitions are delineated on the parameter plane.

Improving linked-lists for neighbor finding with the use of tree search algorithms is proposed here, aiming to cope with highly non-uniform resolution simulations employing a meshless method. The new procedure, coined Quadtree Cells Grid, has been implemented in Smoothed Particle Hydrodynamics (SPH). The SPH scheme employed is adaptive, thus allowing for particle refinement in desired regions of the flow. Owing to the wide range of coexisting particle mass levels, standard linked-list neighbor search algorithms become ineffective. Hence, an alternative is found based on the use of hierarchical data structures, using quadtrees (in 2D problems). The present algorithm exploits the advantages of both linked-lists and quadtree methods with the goal of increasing computational efficiency, when dealing with highly non-uniform particle distributions. Test cases involving two distinct flow problems have demonstrated that the computational cost of the current adaptive neighbor finding algorithm scales linearly with the total number of particles, thus retrieving this characteristic of linked-lists in uniform grid search. Nevertheless, the memory usage increased as a result of the more complex data structure.

This paper presents a high order time discretization method by combining the semi-implicit spectral deferred correction method with energy stable linear schemes to simulate a series of phase field problems. We start with the linear scheme, which is based on the invariant energy quadratization approach and is proved to be linear unconditionally energy stable. The scheme also takes advantage of avoiding nonlinear iteration and the restriction of time step to guarantee the nonlinear system uniquely solvable. Moreover, the scheme leads to linear algebraic system to solve at each iteration, and we employ the multigrid solver to solve it efficiently. Numerical results are given to illustrate that the combination of local discontinuous Galerkin (LDG) spatial discretization and the high order temporal scheme is a practical, accurate and efficient simulation tool when solving phase field problems. Namely, we can obtain high order accuracy in both time and space by solving some simple linear algebraic equations.

Charge transport in suspended monolayer graphene is simulated by a numerical deterministic approach, based on a discontinuous Galerkin (DG) method, for solving the semiclassical Boltzmann equation for electrons. Both the conduction and valence bands are included and the interband scatterings are taken into account.
The use of a Direct Simulation Monte Carlo (DSMC) approach, which properly describes the interband scatterings, is computationally very expensive because the valence band is very populated and a huge number of particles are needed. Also the choice of simulating holes instead of electrons does not overcome the problem because there is a certain degree of ambiguity in the generation and recombination terms of electron-hole pairs. Often, direct solutions of the Boltzmann equations with a DSMC neglect the interband scatterings on the basis of physical arguments. The DG approach does not suffer from the previous drawbacks and requires a reasonable computing effort.
In the present paper the importance of the interband scatterings is accurately evaluated for several values of the Fermi energy, addressing the issue related to the validity of neglecting the generation-recombination terms. It is found out that the inclusion of the interband scatterings produces huge variations in the average values, as the current, with zero Fermi energy while, as expected, the effect of the interband scattering becomes negligible by increasing the absolute value of the Fermi energy.

A nonlinear finite volume element scheme for anisotropic diffusion problems on general triangular meshes is proposed. Starting with a standard linear conforming finite volume element approximation, a corrective term with respect to the flux jumps across element boundaries is added to make the scheme satisfy the discrete maximum principle. The new scheme is free of the anisotropic non-obtuse angle condition which is a severe restriction on the grids for problems with anisotropic diffusion. Moreover, this manipulation can nearly keep the same accuracy as the original scheme. We prove the existence of the numerical solution for this nonlinear scheme theoretically. Numerical results and a grid convergence study are presented for both continuous and discontinuous anisotropic diffusion problems.

This article is devoted to studying the application of the weak Galerkin (WG) finite element method to the elliptic eigenvalue problem with an emphasis on obtaining lower bounds. The WG method uses discontinuous polynomials on polygonal or polyhedral finite element partitions. The non-conforming finite element space of the WG method is the key of the lower bound property. It also makes the WG method more robust and flexible in solving eigenvalue problems. We demonstrate that the WG method can achieve arbitrary high convergence order. This is in contrast with existing nonconforming finite element methods which can provide lower bound approximations by linear finite elements. Numerical results are presented to demonstrate the efficiency and accuracy of the theoretical results.

This paper presents an absorbing boundary condition for molecular dynamics simulations of materials defects. The purpose of the boundary condition is to eliminates spurious reflections of phonons at the boundary and minimize the finite size effect. In contrast to other existing methods, our emphasis is placed on the ease of implementation. In particular, we propose a method for which the implementation can be done within existing molecular dynamics code, and it is insensitive to lattice structure, the geometry and space dimension of the computational domain. To demonstrate the effectiveness, the results from two test problems are presented.

In this paper, we propose a new gradient vector flow model with advection enhancement, called advection-enhanced gradient vector flow, for calculating the external force employed in the active-contour image segmentation. The proposed model is mainly inspired by the functional derivative of an adaptive total variation regularizer whose minimizer is expected to be able to effectively preserve the desired object boundary. More specifically, by incorporating an additional advection term into the usual gradient vector flow model, the resulting external force can much better help the active contour to recover missing edges, to converge to a narrow and deep concavity, and to preserve weak edges. Numerical experiments are performed to demonstrate the high performance of the newly proposed model.

This paper presents a new numerical technique for solving initial and boundary value problems with unsteady strongly nonlinear advection diffusion reaction (ADR) equations. The method is based on the use of the radial basis functions (RBF) for the approximation space of the solution. The Crank-Nicolson scheme is used for approximation in time. This results in a sequence of stationary nonlinear ADR equations. The equations are solved sequentially at each time step using the proposed semi-analytical technique based on the RBFs. The approximate solution is sought in the form of the analytical expansion over basis functions and contains free parameters. The basis functions are constructed in such a way that the expansion satisfies the boundary conditions of the problem for any choice of the free parameters. The free parameters are determined by substitution of the expansion in the equation and collocation in the solution domain. In the case of a nonlinear equation, we use the well-known procedure of quasilinearization. This transforms the original equation into a sequence of the linear ones on each time layer. The numerical examples confirm the high accuracy and robustness of the proposed numerical scheme.

We develop a non-overlapping domain decomposition method (DDM) for scalar wave scattering by periodic layered media. Our approach relies on robust boundary-integral equation formulations of Robin-to-Robin (RtR) maps throughout the frequency spectrum, including cutoff (or Wood) frequencies. We overcome the obstacle of non-convergent quasi-periodic Green functions at these frequencies by incorporating newly introduced shifted Green functions. Using the latter in the definition of quasi-periodic boundary-integral operators leads to rigorously stable computations of RtR operators. We develop Nyström discretizations of the RtR maps that rely on trigonometric interpolation, singularity resolution, and fast convergent windowed quasi-periodic Green functions. We solve the tridiagonal DDM system via recursive Schur complements and establish rigorously that this procedure is always completed successfully. We present a variety of numerical results concerning Wood frequencies in two and three dimensions as well as large numbers of layers.