Several new domain-type and boundary-type numerical discretization schemes with radial basis function

TL;DR

This paper introduces several novel, symmetric, meshless RBF-based numerical schemes for discretizing PDEs, including boundary and domain methods, with improved accuracy and applicability to complex geometries.

Contribution

The paper develops new symmetric boundary and domain RBF discretization schemes, such as the direct and indirect BKM, RBF-based BPM, MKM, and FKM, enhancing accuracy and efficiency in PDE solutions.

Findings

Symmetric boundary knot methods are effective regardless of boundary geometry.

The modified Kansa method reduces boundary node errors.

The schemes are meshless, integration-free, and produce sparse matrices.

Abstract

This paper is concerned with a few novel RBF-based numerical schemes discretizing partial differential equations. For boundary-type methods, we derive the indirect and direct symmetric boundary knot methods (BKM). The resulting interpolation matrix of both is always symmetric irrespective of boundary geometry and conditions. In particular, the direct BKM applies the practical physical variables rather than expansion coefficients and becomes very competitive to the boundary element method. On the other hand, based on the multiple reciprocity principle, we invent the RBF-based boundary particle method (BPM) for general inhomogeneous problems without a need using inner nodes. The direct and symmetric BPM schemes are also developed. For domain-type RBF discretization schemes, by using the Green integral we develop a new Hermite RBF scheme called as the modified Kansa method (MKM), which differs from the symmetric Hermite RBF scheme in that the MKM discretizes both governing equation and boundary conditions on the same boundary nodes. The local spline version of the MKM is named as the finite knot method (FKM). Both MKM and FKM significantly reduce calculation errors at nodes adjacent to boundary. In addition, the nonsingular high-order fundamental or general solution is strongly recommended as the RBF in the domain-type methods and dual reciprocity method approximation of particular solution relating to the BKM. It is stressed that all the above discretization methods of boundary-type and domain-type are symmetric, meshless, and integration-free. The spline-based schemes will produce desirable symmetric sparse banded interpolation matrix. In appendix, we present a Hermite scheme to eliminate edge effect on the RBF geometric modeling and imaging.