Incremental Unknowns for the Heat Equation with Time-Dependent Coefficients: Semi-Implicit $\theta$-Schemes and Their Stability

Abstract

Based on the finite difference discretization of partial differential equations, we propose a kind of semi-implicit $\theta$-schemes of incremental unknowns type for the heat equation with time-dependent coefficients. The stability of the new schemes is carefully studied. Some new types of conditions give better stability when $\theta$ is closed to $1/2$ even if we have variable coefficients.