BACKWARD EULER FINITE DIFFERENCE APPROXIMATIONS OF A LOGARITHMIC HEAT EQUATION OVER A 2D RECTANGULAR DOMAIN

Panagiotis Paraschis (1) , Georgios E Zouraris (2)

1 School of Applied Mathematical and Physical Sciences, National Technical University of Athens
2 Department of Mathematics and Applied Mathematics, University of Crete (Heraklion, Greece)

We formulate an initial and Dirichlet boundary value problem for a semilinear heat equation with logarithmic nonlinearity over a two dimensional rectangular domain. We approximate its solution by using for space discretization the standard second order finite difference scheme, and for time-stepping the Linearized Backward Euler method, or, the ε−Backward Euler method after applying a smooth ε−cutoff of the logarithmic term, where the small positive parameter ε acts as a discretization parameter (along with the time-step and the space mesh widths) and has no influence on the complexity of the method. We prove optimal order error estimates in the discrete L ∞ t (L 2 x) and the discrete L ∞ t (L ∞ x) norm, where the constants are ε−free and no mesh conditions are imposed. Results from numerical experiments expose the efficiency of the numerical methods proposed. It is the first time in the literature where numerical methods for the approximation of the solution to the heat equation with logarithmic nonlinearity are applied and analysed.

Domaines: Mathématiques [math]

Panagiotis Paraschis, Georgios E Zouraris. BACKWARD EULER FINITE DIFFERENCE APPROXIMATIONS OF A LOGARITHMIC HEAT EQUATION OVER A 2D RECTANGULAR DOMAIN. 2021. ⟨hal-03220015⟩