A fictitious domain spectral method for time-dependent Sobolev equations in complex domains

Numerical simulation of time-dependent Sobolev-type equations in complex domains remains challenging, primarily due to the interplay of high-order derivatives, nonlinearities, and irregular boundaries. While spectral methods offer high accuracy, their application is largely confined to regular domains. To overcome this limitation, we propose an efficient spectral framework based on a fictitious domain formulation. The physical domain is embedded into a larger, circular auxiliary domain. A polar coordinate transformation is employed, enabling discretization via a Fourier spectral method in the angular direction. This reduces the problem to a set of one-dimensional radial subproblems, which are solved with high accuracy using a Legendre spectral-Galerkin method. For time integration, high-order Backward Differentiation Formula (BDFk) and Implicit–Explicit (IMEX) schemes are applied to the linear and nonlinear terms, respectively. Numerical experiments demonstrate the method’s spectral accuracy, robustness, and effectiveness across diverse challenging cases, including smooth solutions and those with singularities or sharp gradients. This work provides a flexible and powerful computational paradigm for simulating complex transient phenomena governed by Sobolev-type equations.