A Bernoulli-Barycentric Rational Matrix Collocation Method With Preconditioning for a Class of Evolutionary PDEs

We propose a Bernoulli-barycentric rational matrix collocation method for two-dimensional evolutionary partial differential equations (PDEs) with variable coefficients. This method absorbs Bernoulli polynomials and barycentric rational interpolations as the basis functions in time and space, respectively. The theoretical accuracy of the proposed numerical scheme is proven to be (Formula presented.), where (Formula presented.) is the number of basis functions in time, and (Formula presented.) and (Formula presented.) are the grid sizes in the (Formula presented.) and (Formula presented.) directions, respectively. Additionally, (Formula presented.). To efficiently solve the linear systems arising from the discretizations, we introduce a class of dimension-expanded preconditioners that leverage the structural properties of the coefficient matrices. A theoretical analysis of the eigenvalue distributions of the preconditioned matrices is provided. The effectiveness of the proposed method and preconditioners is demonstrated through numerical experiments on real-world examples, including the heat conduction equation, the advection-diffusion equation, the wave equation, and telegraph equations.