30 research outputs found
A kernel-based meshless conservative Galerkin method for solving Hamiltonian wave equations
We propose a meshless conservative Galerkin method for solving Hamiltonian
wave equations. We first discretize the equation in space using radial basis
functions in a Galerkin-type formulation. Differ from the traditional RBF
Galerkin method that directly uses nonlinear functions in its weak form, our
method employs appropriate projection operators in the construction of the
Galerkin equation, which will be shown to conserve global energies. Moreover,
we provide a complete error analysis to the proposed discretization. We further
derive the fully discretized solution by a second order average vector field
scheme. We prove that the fully discretized solution preserved the discretized
energy exactly. Finally, we provide some numerical examples to demonstrate the
accuracy and the energy conservation
Adaptive radial basis function generated finite-difference (RBF-FD) on non-uniform nodes using -refinement
Radial basis functions-generated finite difference methods (RBF-FDs) have
been gaining popularity recently. In particular, the RBF-FD based on
polyharmonic splines (PHS) augmented with multivariate polynomials (PHS+poly)
has been found significantly effective. For the approximation order of RBF-FDs'
weights on scattered nodes, one can already find mathematical theories in the
literature. Many practical problems in numerical analysis, however, do not
require a uniform node-distribution. Instead, they would be better suited if
specific areas of the domain, where complicated physics needed to be resolved,
had a relatively higher node-density compared to the rest of the domain. In
this work, we proposed a practical adaptive RBF-FD with a user-defined order of
convergence with respect to the total number of (possibly scattered and
non-uniform) data points . Our algorithm outputs a sparse differentiation
matrix with the desired approximation order. Numerical examples are provided to
show that the proposed adaptive RBF-FD method yields the expected
-convergence even for highly non-uniform node-distributions. The proposed
method also reduces the number of non-zero elements in the linear system
without sacrificing accuracy.Comment: An updated version with seismic modeling will be included in version
Solving moving-boundary problems with the wavelet adaptive radial basis functions method
Moving boundaries are associated with the time-dependent problems where the momentary position of boundaries needs to be determined as a function of time. The level set method has become an effective tool for tracking, modelling and simulating the motion of free boundaries in fluid mechanics, computer animation and image processing. This work extends our earlier work on solving moving boundary problems with adaptive meshless methods. In particular, the objective of this paper is to investigate numerical performance the radial basis functions (RBFs) methods, with compactly supported basis and with global basis, coupled with a wavelet node refinement technique and a greedy trial space selection technique. Numerical simulations are provided to verify the effectiveness and robustness of RBFs methods with different adaptive techniques
Adaptive multiquadric collocation for boundary layer problems
AbstractAn adaptive collocation method based upon radial basis functions is presented for the solution of singularly perturbed two-point boundary value problems. Using a multiquadric integral formulation, the second derivative of the solution is approximated by multiquadric radial basis functions. This approach is combined with a coordinate stretching technique. The required variable transformation is accomplished by a conformal mapping, an iterated sine-transformation. A new error indicator function accurately captures the regions of the interval with insufficient resolution. This indicator is used to adaptively add data centres and collocation points. The method resolves extremely thin layers accurately with fairly few basis functions. The proposed adaptive scheme is very robust, and reaches high accuracy even when parameters in our coordinate stretching technique are not chosen optimally. The effectiveness of our new method is demonstrated on two examples with boundary layers, and one example featuring an interior layer. It is shown in detail how the adaptive method refines the resolution