Adaptive rational spectral methods for the linear stability analysis of nonlinear fourth-order problems

Abstract

This paper presents the application of adaptive rational spectral methods to the linear stability analysis of nonlinear fourth-order problems. Our model equation is a phase-field model of infiltration, but the proposed discretization can be directly extended to similar equations arising in thin film flows. The sharpness and structure of the wetting front preclude the use of the standard Chebyshev pseudo-spectral method, due to its slow convergence in problems where the solution has steep internal layers. We discuss the effectiveness and conditioning of the proposed discretization, and show that it allows the computation of accurate traveling waves and eigenvalues for small values of the initial water saturation/film precursor, several orders of magnitude smaller than the values considered previously in analogous stability analyses of thin film flows, using just a few hundred grid points.

Three-dimensional simulation of unstable gravity-driven infiltration of water into a porous medium

“…We observe that even this small-scale feature of the solution is in agreement for both computations, which provides evidence for our three-dimensional code being correct. 10 We also note that the adaptive rational pseudospectral algorithm, which adaptively selects the interpolation points, and has been shown to vastly outperform standard Chebyshev collocation methods and higher-order finite differences [28], required a minimum step size of h min ≈ 0.005151. This mesh renders a solutions with a relative error of at least 5 · 10 −4 .…”

“…Exact solutions to the infiltration theory (17) are not available, so we will utilize our three-dimensional code to produce one-dimensional traveling wave solutions to the model. The traveling wave solutions can be obtained to high precision using an overkill solution produced by a rational pseudospectral method with adaptively transformed Chebyshev nodes [75,28].…”

“…Computational phase-field modeling has been traditionally dominated by collocation methods, such as finite differences [36,74,25] or pseudo-spectral algorithms [57,60,81,28], by finite volume schemes [27], and by mixed finite element methods [3, 10,31,35]. Since our model equation involves fourth-order partial derivatives, conforming finite element formulations require C 1 continuity of the basis functions across ele-30 ment boundaries.…”

Recent advances in linear barycentric rational interpolation

“…However, when one may choose the nodes, the second linear rational interpolant introduced in [2], when combined with nodes which are conformal maps of Chebyshev points, provides a very efficient means (in the sense that it is as simple as the polynomial interpolant and exponential convergence occurs) of solving various kinds of differential equations with steep solutions [3][4][5][6].…”

Reply to comment by David A. DiCarlo on “A phase field model of unsaturated flow”

“…In a recent series of papers [ Cueto‐Felgueroso and Juanes , 2008, 2009a, 2009b, 2009c], we put forward a framework for modeling multiphase flow in porous media, which recognizes the presence of macroscopic diffuse interfaces (displacement fronts). The proposed framework is based on a phase field approach, which originated in the 1950s in the field of solidification [ Cahn and Hilliard , 1958].…”