A reconstruction-based Chebyshev-collocation method for the Poisson equation: An accurate treatment of the Gibbs-Wilbraham phenomenon on irregular interfaces

Abstract

An accurate numerical solution of the Laplace equation and Poisson equation is desirable in physical problems related to fluid flows, and spectral methods are well suited towards this goal. However, in the presence of an interface, the solution is discontinuous owing to the interface jump conditions and thus the numerical approximation is susceptible to the Gibbs-Wilbraham phenomenon, leading to a reduced solution accuracy. In order to resolve the discontinuity at the interface, we propose a reconstruction technique wherein the approximate solution is expressed as the sum of an infinitely-differentiable smooth function and a modified Heaviside function. The smooth function is composed of Chebyshev polynomials while the modified Heaviside function is expressed in a weak form using the jump conditions across the interface. The reconstruction framework allows us to impose the interfacial conditions exactly. To motivate further application of the technique, we first present the underlying rationale by revisiting the work of Wilbraham (1848) [13], followed by the numerical implementation. We present numerical solution of four equations to illustrate the application of the proposed technique: (i) an ordinary differential equation (ODE), (ii) the Euler-Bernoulli equation; (iii) the Laplace equation and (iv) the Poisson equation. The Laplace and Poisson equation are solved on a square domain embedded with complex interfacial geometries in two-dimensions and on a cube with a spherical interface in three-dimensions. The method achieves error in the maximum norm ∼O(10−14) using approximately 30 modes in each direction and requires a computational time of less than a second for all the problems. We also conduct a sensitivity analysis to demonstrate the robustness of the algorithm under sub-grid scale perturbations to interface location. An alternate derivation for the modified Heaviside function is proposed that is distinct from the Taylor series expansion used to treat interface conditions in difference methods. The pointwise error plots are subsequently presented to demonstrate that the Gibbs-Wilbraham phenomenon is accurately resolved. Finally, we present concluding thoughts on the potential applications of the reconstruction technique.