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Register a new userTowards Stable Radial Basis Function Methods for Linear Advection Problems
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Added by
Jan Glaubitz
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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In this work, we investigate (energy) stability of global radial basis function (RBF) methods for linear advection problems. Classically, boundary conditions (BC) are enforced strongly in RBF methods. By now it is well-known that this can lead to stability problems, however. Here, we follow a different path and propose two novel RBF approaches which are based on a weak enforcement of BCs. By using the concept of flux reconstruction and simultaneous approximation terms (SATs), respectively, we are able to prove that both new RBF schemes are strongly (energy) stable. Numerical results in one and two spatial dimensions for both scalar equations and systems are presented, supporting our theoretical analysis.
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It is well understood that boundary conditions (BCs) may cause global radial basis function (RBF) methods to become unstable for hyperbolic conservation laws (CLs). Here we investigate this phenomenon and identify the strong enforcement of BCs as the mechanism triggering such stability issues. Based on this observation we propose a technique to weakly enforce BCs in RBF methods. In the case of hyperbolic CLs, this is achieved by carefully building RBF methods from the weak form of the CL, rather than the typically enforced strong form. Furthermore, we demonstrate that global RBF methods may violate conservation, yielding physically unreasonable solutions when the approximation does not take into account these considerations. Numerical experiments validate our theoretical results.
Meshfree methods based on radial basis function (RBF) approximation are of interest for numerical solution of partial differential equations (PDEs) because they are flexible with respect to the geometry of the computational domain, they can provide high order convergence, they are not more complicated for problems with many space dimensions and they allow for local refinement. The aim of this paper is to show that the solution of the Rosenau equation, as an example of an initial-boundary value problem with multiple boundary conditions, can be implemented using RBF approximation methods. We extend the fictitious point method and the resampling method to work in combination with an RBF collocation method. Both approaches are implemented in one and two space dimensions. The accuracy of the RBF fictitious point method is analysed partly theoretically and partly numerically. The error estimates indicate that a high order of convergence can be achieved for the Rosenau equation. The numerical experiments show that both methods perform well. In the one-dimensional case, the accuracy of the RBF approaches is compared with that of a pseudospectral resampling method, showing similar or slightly better accuracy for the RBF methods. In the two-dimensional case, the Rosenau problem is solved both in a square domain and in a starfish-shaped domain, to illustrate the capability of the RBF-based methods to handle irregular geometries.
This paper addresses the question whether there are numerical schemes for constant-coefficient advection problems that can yield convergent solutions for an infinite time horizon. The motivation is that such methods may serve as building blocks for long-time accurate solutions in more complex advection-dominated problems. After establishing a new notion of convergence in an infinite time limit of numerical methods, we first show that linear methods cannot meet this convergence criterion. Then we present a new numerical methodology, based on a nonlinear jet scheme framework. We show that these methods do satisfy the new convergence criterion, thus establishing that numerical methods exist that converge on an infinite time horizon, and demonstrate the long-time accuracy gains incurred by this property.
We provide a preliminary comparison of the dispersion properties, specifically the time-amplification factor, the scaled group velocity and the error in the phase speed of four spatiotemporal discretization schemes utilized for solving the one-dimensional (1D) linear advection diffusion reaction (ADR) equation: (a) An explicit (RK2) temporal integration combined with the Optimal Upwind Compact Scheme (or OUCS3) and the central difference scheme (CD2) for second order spatial discretization, (b) a fully implicit mid-point rule for time integration coupled with the OUCS3 and the Leles compact scheme for first and second order spatial discretization, respectively, (c) An implicit (mid-point rule)-explicit (RK2) or IMEX time integration blended with OUCS3 and Leles compact scheme (where the IMEX time integration follows the same ideology as introduced by Ascher et al.), and (d) the IMEX (mid-point/RK2) time integration melded with the New Combined Compact Difference scheme (or NCCD scheme). Analysis reveal the superior resolution features of the IMEX-NCCD scheme including an enhanced region of neutral stability (a region where the amplification factor is close to one), a diminished region of spurious propagation characteristics (or a region of negative group velocity) and a smaller region of nonzero phase speed error. The dispersion error of these numerical schemes through the role of q-waves is further investigated using the novel error propagation equation for the 1D linear ADR equation. Again, the in silico experiments divulge excellent Dispersion Relation Preservation (DRP) properties of the IMEX-NCCD scheme including minimal dissipation via implicit filtering and negligible unphysical oscillations (or Gibbs phenomena) on coarser grids.
Localized collocation methods based on radial basis functions (RBFs) for elliptic problems appear to be non-robust in the presence of Neumann boundary conditions. In this paper we overcome this issue by formulating the RBF-generated finite difference method in a discrete least-squares setting instead. This allows us to prove high-order convergence under node refinement and to numerically verify that the least-squares formulation is more accurate and robust than the collocation formulation. The implementation effort for the modified algorithm is comparable to that for the collocation method.
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