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We describe a new approach to derive numerical approximations of boundary conditions for high-order accurate finite-difference approximations. The approach, called the local compatibility boundary condition (LCBC) method, uses boundary conditions and compatibility boundary conditions derived from the governing equations, as well as interior and boundary grid values, to construct a local polynomial, whose degree matches the order of accuracy of the interior scheme, centered at each boundary point. The local polynomial is then used to derive a discrete formula for each ghost point in terms of the data. This approach leads to centered approximations that are generally more accurate and stable than one-sided approximations. Moreover, the stencil approximations are local since they do not couple to neighboring ghost-point values, which can occur with traditional compatibility conditions. The local polynomial is derived using continuous operators and derivatives, which enables the automatic construction of stencil approximations at different orders of accuracy. The LCBC method is developed here for problems governed by second-order partial differential equations, and it is verified in two space dimensions for schemes up to sixth-order accuracy. [ABSTRACT FROM AUTHOR]

This manuscript discusses discretization of the Vlasov-Poisson system in 2D+2V phase space using high-order accurate conservative finite difference algorithms. One challenge confronting direct kinetic simulation is the significant computational cost associated with the high-dimensional phase space description. In the present work we advocate the use of high-order accurate schemes as a mechanism to reduce the computational cost required to deliver a given level of error in the computed solution. We pursue a discretely conservative finite difference formulation of the governing equations, and discuss fourth- and sixth-order accurate schemes. In addition, we employ a minimally dissipative nonlinear scheme based on the well-known WENO (weighted essentially nonoscillatory) approach. Verification of the full formulation is performed using the method of manufactured solutions. Results are also presented for the physically relevant scenarios of Landau damping, and growth of transverse instabilities from an imposed plane wave. [ABSTRACT FROM AUTHOR]

An analysis is made of a new partitioned scheme for solving fluid-structure interaction problems involving viscous incompressible flow and compressible elastic-solids. The new scheme is stable, without sub-time-step iterations, even for light solids when added-mass and addeddamping effects are large. The fluid is updated with an implicit-explicit (IMEX) fractional-step scheme whereby the velocity is advanced in one step, treating the viscous terms implicitly, and the pressure is computed in a second step. The key components of the scheme are a Robin (mixed) interface condition for the fluid pressure, and impedance-based interface conditions for the velocity. While the impedance for the solid is well defined, the fluid impedance is not, and a semidiscrete local analysis is used to inform this choice. The properties of the new scheme are analyzed, and numerical results are presented to confirm the stability and accuracy of the scheme. [ABSTRACT FROM AUTHOR]

The concentration of ions, or osmolarity, in the tear film is a key variable in understanding dry eye symptoms and disease. In this manuscript, we derive a mathematical model that couples osmolarity (treated as a single solute) and fluid dynamics within the tear film on a 2D eye-shaped domain. The model includes the physical effects of evaporation, surface tension, viscosity, ocular surface wettability, osmolarity, osmosis and tear fluid supply and drainage. The governing system of coupled non-linear partial differential equations is solved using the Overture computational framework, together with a hybrid time-stepping scheme, using a variable step backward differentiation formula and a Runge-Kutta-Chebyshev method that were added to the framework. The results of our numerical simulations provide new insight into the osmolarity distribution over the ocular surface during the interblink. [ABSTRACT FROM AUTHOR]

The bunching instability of particles trapped in Langmuir waves is studied using Vlasov simulations. A measure of particle bunching is defined and used to extract the growth rate from numerical simulations, which are compared with theory [Dodin et al., Phys. Rev. Lett. 110, 215006 (2013)]. In addition, the general theory of trapped particle instability in 1D is revisited and a more accurate description of the dispersion relation is obtained. Excellent agreement between numerical and theoretical predictions of growth rates of the bunching instability is shown over a range of parameters. [ABSTRACT FROM AUTHOR]