Wavelet-based grid adaptation with consistent treatment of high-order sharp immersed geometries

Abstract

Wavelet-based grid adaptation methods use multiresolution analysis for error estimation, offering a mathematically rigorous approach to adaptive grid refinement when solving Partial Differential Equations (PDEs). However, applying these methods to PDE discretizations with immersed geometries is challenging, as standard interpolating wavelet transforms lose consistency near non-grid-aligned boundary intersections. To address this, we propose a high-order interpolating wavelet transform adaptation strategy compatible with sharp immersed boundary and interface discretizations. The approach performs consistent high-order wavelet transforms on narrow intervals using a 1D polynomial extrapolation technique. To maintain high order, the technique incorporates boundary values and derivatives, which are evaluated from multivariate interpolating polynomials similar to those used in high order immersed finite difference discretizations. Consequently, the proposed approach maintains the wavelet order on any arbitrary smooth multidimensional domain, including near concave geometry sections. This approach enables grid adaptation in complex domains while robustly bounding the numerical error via a manually set refinement threshold. The algorithm's performance is validated on both static and dynamic problems, including the Navier-Stokes equations with moving boundaries and temporally adapting grid resolutions. The results demonstrate that the proposed method enables effective grid adaptation, establishing a robust, predictable relationship between a user-defined refinement threshold and the overall solution error, even for problems with complex, moving boundaries.

Figures (13)

• Figure 1: Left: Three types of extrapolation strategies in the proposed wavelet transform algorithm. For a sufficient number of grid points, when the immediate neighbor of the control point has an even (odd) index, Type I (Type II) extrapolation is used. For an insufficient number of grid points (concave sections in $>1$D), a Hermite-like extrapolation is used. Right: Modified scaling functions near the boundary for Type I (blue) and Type II (red) extrapolations, for different wavelet orders $N$ (top to bottom).
• Figure 2: Left: When the boundary is convex (top, red line) a 'wide interval' wavelet transform can be performed. For a concave section (bottom, green) there might be insufficient grid points to apply the 1D extrapolation stencil along a grid line. Right: when such a concavity occurs, we employ the half-elliptical least square fit strategy. For each control point elliptical stencils are used to construct polynomial approximations to the high order derivatives on the boundary. These derivatives are in turn used to perform a Hermite-like 1D polynomial extrapolation along the grid line, which enables the 1D wavelet transform on the small interval between the two control points.
• Figure 3: Errors, compression rates, and detail coefficient scaling for a static compression test of a smooth field with an immersed non-convex geometry, using lifted ($N.2$) and non-lifted ($N.0$) wavelets of varying order $N$.
• Figure 4: A (6, 2) order wavelet transform of a static field around a star shaped embedded geometry. The scaling and detail coefficients are plotted in log scale. The detail coefficients $|\gamma_x|$ and $|\gamma_y|$ have slightly larger magnitude near the star surface, due to the boundary amplification effect. $|\gamma_{xy}|$ decays as $\mathcal{O}(h^{2N})$ in free space, and $\mathcal{O}(h^N)$ near the boundary.
• Figure 5: Left: Reference solution to the diffusion problem of Section 5.2 at t=5.0; Right: Pointwise numerical error around the top-left of the geometry as a function of refinement threshold $\varepsilon_r$.

Theorems & Definitions (3)

• proposition 1
• proposition 2
• proposition 3