Overview
A nonstandard finite difference (NSFD) scheme has been developed and analyzed for nonlinear parabolic equations that incorporate a p-Laplacian-type diffusion operator. This scheme is applicable in both one- and two-dimensional spatial domains. The design follows Mickens' principles, utilizing a nonlinear denominator function $\phi(.)$ and a nonlocal approximation for the nonlinear diffusion term $\Delta_p$. This approach aims to create a discrete model that preserves key structural properties of the continuous problem.
Research Context
The research focuses on addressing challenges associated with nonlinear parabolic equations involving p-Laplacian-type diffusion. Standard finite difference methods (FDMs) can sometimes lead to a loss of qualitative properties inherent in the continuous problem, such as positivity, boundedness, and stability. The development of the NSFD scheme was motivated by the need for a discretization method that could retain these properties.
Approach
The NSFD scheme was formulated using Mickens' design principles. A central element of this design is the incorporation of a nonlinear denominator function, denoted as $\phi(.)$. Additionally, a nonlocal approximation was employed for the nonlinear diffusion term, $\Delta_p$. This combination results in a discrete model engineered to preserve specific qualitative characteristics of the continuous problem. The research involved establishing the well-posedness of the continuous model, deriving the NSFD scheme, and subsequently investigating its consistency, convergence, and local truncation error.
Findings
- The proposed NSFD scheme is designed to retain key qualitative properties of the continuous problem, specifically positivity, boundedness, and stability. These properties may be lost when using standard finite difference methods.
- The scheme employs a nonlinear denominator function $\phi(.)$ and a nonlocal approximation of the nonlinear diffusion term $\Delta_p$, resulting in a structure-preserving discrete model.
- The well-posedness of the continuous model was established.
- The NSFD scheme's consistency, convergence, and local truncation error were investigated.
- Numerical experiments confirmed the theoretical results.
- Unlike the standard explicit FDM, the proposed NSFD scheme avoids spurious oscillations.
- The NSFD scheme prevents nonphysical negative solutions, even when relatively large time-step sizes are used.
Why This Matters
This scheme offers a method for numerically solving a class of nonlinear parabolic equations while preserving fundamental physical properties like positivity and stability. This preservation is crucial for obtaining reliable simulation results, particularly in scenarios where standard methods might introduce nonphysical artifacts or instabilities, even with larger time steps.