Compact LABFM: a framework for meshless methods with spectral-like resolving power

Abstract

Meshless methods are often used in numerical simulations of systems of partial differential equations (PDEs), particularly those which involve complex geometries or free surfaces. Here we present a novel compact scheme based on the local anisotropic basis function method (LABFM), a meshless method which provides approximations to spatial operators to arbitrary polynomial consistency. Our approach mimics compact finite-differences by using implicit stencils to optimise the resolving power of each operator, whilst retaining diagonal dominance of the resulting global sparse linear system. The new method is demonstrated to provide improved approximations by a series of convergence tests and resolving power analysis, before solutions to canonical PDEs are computed. Significant gains in accuracy are observed, in particular for solutions containing high wavenumber components. Our compact meshfree method provides a pathway to high-order simulations of PDEs in complex geometries with spectral-like resolving power, and has the potential to lead to a step-change in the accuracy of numerical solutions to such problems.

Figures (14)

• Figure 1: An illustrative example of how an implicit stencil is chosen from within the right-hand stencil for compact LABFM, with $Q_i=5$. (a) Gradient Operator with respect to $x$; (b) Gradient Operator with respect to $y$; (c) Laplacian Operator. Solid colour nodes correspond to those in the implicit stencil.
• Figure 2: Real part of resolving power of the gradient operator for second order accurate (blue lines) and fourth order (red lines) accurate operators, with schemes as labelled in Table \ref{['tab:stencils']} except for the spectral scheme (i) (exact differentiation). Left Panel $k_y=0$, Centre Panel $k_y=k_x$, Right Panel $k_y=2k_x$.
• Figure 3: Error $\varepsilon_1$ of real part of resolving power of the gradient operator for second order accurate (blue lines) and fourth order (red lines) accurate operators, with schemes as labelled in Table \ref{['tab:stencils']}. Left Panel $k_y=0$, Centre Panel $k_y=k_x$, Right Panel $k_y=2k_x$.
• Figure 4: Imaginary part of the resolving power of the gradient operator for second order accurate (blue lines) and fourth order (red lines) accurate operators, with schemes as labelled in Table \ref{['tab:stencils']}. Left Panel $k_y=0$, Centre Panel $k_y=k_x$, Right Panel $k_y=2k_x$.
• Figure 5: Resolving power of the Laplacian operator for second order accurate (blue lines) and fourth order (red lines) accurate operators, with schemes as labelled in Table \ref{['tab:stencils']} except for the spectral scheme (i) (exact differentiation). Left Panel $k_y=0$, Centre Panel $k_y=k_x$, Right Panel $k_x=0$.