High order locally one-dimensional methods for solving two-dimensional parabolic equations

Abstract

Based on the locally one-dimensional strategy, we propose two high order finite difference schemes for solving two-dimensional linear parabolic equations. In the first method, fourth order approximation in space and (2, 2) Padé formula in time are considered. These lead to a fourth order finite difference scheme in both space and time. For the second method, we employ sixth order approximation in space and (3, 3) Padé formula in time. This yields a novel sixth order scheme in both space and time. The methods are proved to be unconditionally stable, and the Sheng-Suzuki barrier is successfully avoided. Numerical experiments are given to illustrate our conclusions as well as computational effectiveness.

Modeling and Monte Carlo simulation on photothermal effect in Gd3Al3Ga2O12:Ce3+/Y3Al5O12:Cr3+ layered composite ceramic

“…The two‐dimensional heat conduction equation of layered composite ceramics is established. The two‐dimensional thermal conduction equation is solved by difference method to obtain the approximate solution of each discrete point 38,39 . The effect of light source characteristics on the temperature distribution in layered ceramics was studied.…”

“…The two-dimensional thermal conduction equation is solved by difference method to obtain the approximate solution of each discrete point. 38,39 The effect of light source characteristics on the temperature distribution in layered ceramics was studied. The light absorption and the distribution of temperature field in the whole space were compared.…”

Hermite Interpolation Approach to High-Order Approximation of Heat Equations

“…Compact scheme is used in solving coupled Burgers' equation in [6]. Also, [7] proposes two high order finite difference schemes for solving twodimensional linear parabolic equations where fourth order compact schemes were considered. Also, [8] derived the compact theta scheme for the linear initial-boundary value problems, while [9] formulated a fourth order compact scheme that satisfies the Lie symmetry.…”

A new numerical scheme for solving the two dimensional fractional diffusion equation

“…In this method, the alternating directions implicit (ADI) approach, the Crank-Nicolson discretization and the Richardson extrapolation are used [10]. In [11] the Grünwald finite difference approximation is used for the fractional derivative with ADI method and locally one dimensional (LOD) method is used in [12][13][14][15][16]. In this study, we consider the two dimensional time fractional diffusion equation of the form [9] , , =…”