We formulate an oversampled radial basis function generated finite difference (RBF-FD) method to solve time-dependent nonlinear conservation laws. The analytic solutions of these problems are known to be discontinuous, which leads to occurrence of no
n-physical oscillations (Gibbs phenomenon) that pollute the numerical solutions and can make them unstable. We address these difficulties using a residual based artificial viscosity stabilization, where the residual of the conservation law indicates the approximate location of the shocks. The location is then used to locally apply an upwind viscosity term, which stabilizes the Gibbs phenomenon and does not smear the solution away from the shocks. The proposed method is numerically tested and proves to be robust and accurate when solving scalar conservation laws and systems of conservation laws, such as compressible Euler equations.
We introduce a hybrid method to couple continuous Galerkin finite element methods and high-order finite difference methods in a nonconforming multiblock fashion. The aim is to optimize computational efficiency when complex geometries are present. The
proposed coupling technique requires minimal changes in the existing schemes while maintaining strict stability, accuracy, and energy conservation. Results are demonstrated on linear and nonlinear scalar conservation laws in two spatial dimensions.
This paper discusses some regularity of almost periodic solutions of the Poissons equation $-Delta u = f$ in $mathbb{R}^n$, where $f$ is an almost periodic function. It has been proved by Sibuya [Almost periodic solutions of Poissons equation. Proc.
Amer. Math. Soc., 28:195--198, 1971.] that if $u$ is a bounded continuous function and solves the Poissons equation in the distribution sense, then $u$ is an almost periodic function. In this work, we relax the assumption of the usual boundedness into boundedness in the sense of distribution which we refer to as a bounded generalized function. The set of bounded generalized functions are wider than the set of usual bounded functions. Then, upon assuming that $u$ is a bounded generalized function and solves the Poissons equation in the distribution sense, we prove that this solution is bounded in the usual sense, continuous and almost periodic. Moreover, we show that the first partial derivatives of the solution $partial u/ partial x_i$, $i=1, ldots, n$, are also continuous, bounded, and almost periodic functions. The technique is based on extending a representation formula using Greens function for Poissons equation for solutions in the distribution sense. Some useful properties of distributions are also shown that can be used to study other elliptic problems.
We propose a novel algorithm for the approximation of surface-quasi geostrophic (SQG) flows modeled by a nonlinear partial differential equation coupling transport and fractional diffusion phenomena. The time discretization consists of an explicit st
rong-stability-preserving three-stage Runge-Kutta method while a flux-corrected-transport (FCT) method coupled with Dunford-Taylor representations of fractional operators is advocated for the space discretization. Standard continuous piecewise linear finite elements are employed and the algorithm does not have restrictions on the mesh structure nor on the computational domain. In the inviscid case, we show that the resulting scheme satisfies a discrete maximum principle property under a standard CFL condition and observe, in practice, its second-order accuracy in space. The algorithm successfully approximates several benchmarks with sharp transitions and fine structures typical of SQG flows. In addition, theoretical Kolmogorov energy decay rates are observed on a freely decaying atmospheric turbulence simulation.
A new second-order method for approximating the compressible Euler equations is introduced. The method preserves all the known invariant domains of the Euler system: positivity of the density, positivity of the internal energy and the local minimum p
rinciple on the specific entropy. The technique combines a first-order, invariant domain preserving, Guaranteed Maximum Speed method using a Graph Viscosity (GMS-GV1) with an invariant domain violating, but entropy consistent, high-order method. Invariant domain preserving auxiliary states, naturally produced by the GMS-GV1 method, are used to define local bounds for the high-order method which is then made invariant domain preserving via a convex limiting process. Numerical tests confirm the second-order accuracy of the new GMS-GV2 method in the maximum norm, where 2 stands for second-order. The proposed convex limiting is generic and can be applied to other approximation techniques and other hyperbolic systems.
