Two‐Level Implicit High Accuracy Method in Exponential Form for 2D Quasi‐Linear Parabolic Equations on Irrational Domain

ABSTRACT This paper presents a novel compact implicit method in exponential form for solving two‐dimensional quasi‐linear parabolic partial differential equations on an irrational domain using an unequal mesh. The method's spatial accuracy is fourth order, and its temporal accuracy is second order. The compact scheme is designed to work on an irrational domain, utilizing only two compact cells, each comprising 9 grid points. Additionally, we derive an unconditionally stable alternating direction implicit method for the heat conduction equation in polar coordinates. The generalization of the proposed scheme to the system of equations is also shown, and the performance of the extended scheme is thoroughly evaluated for solving 2D linear fourth‐order parabolic equations. Its numerical results and accuracy are validated through a series of problems, including the Burgers' equation, the heat conduction equation in polar coordinates, the Navier–Stokes equations, the quasi‐linear parabolic problem, the extended Fisher–Kolmogorov equation, the Taylor–Vortex problem, and the Kuramoto–Sivashinsky equation.