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A fourth-order arithmetic average compact finite-difference method for nonlinear singular elliptic PDEs on a 3D
1Faculty of Mathematics and Computer Science, South Asian University, Maidan Garhi, New Delhi 110 068, India.
Methodsx
|October 17, 2023
Summary
A new numerical method using compact discretization on a quasi-variable grid effectively solves nonlinear 3D elliptic PDEs common in battery models. This approach offers accurate solutions and fourth-order convergence, even for singular models.
Area of Science:
- Numerical Analysis
- Computational Physics
- Applied Mathematics
Background:
- Nonlinear elliptic partial differential equations (PDEs) are crucial for modeling phenomena like convection-dominated diffusion in batteries.
- Closed-form solutions for these complex models are often unavailable, necessitating robust numerical methods.
- Existing numerical treatments can struggle with singular models and error localization.
Purpose of the Study:
- To introduce a novel numerical method for solving a broad class of nonlinear three-dimensional elliptic PDEs.
- To address the limitations of traditional methods in handling singular models and error dispersion.
- To provide a reliable approach for analyzing the long-term and quantitative behavior of battery diffusion models.
Main Methods:
- Development of an arithmetic average compact discretization scheme.
- Implementation on a quasi-variable grid network, requiring only nineteen-point grids.
- Analysis of truncation error dissemination and grid stretching parameter effects.
- Examination of convergence properties using monotone and irreducible matrices.
Main Results:
- The proposed method demonstrates applicability to singular nonlinear elliptic PDEs.
- It effectively disperses truncation errors across the domain, unlike fixed-step methods.
- Numerical analysis confirmed fourth-order convergence for various 3D elliptic PDEs.
- Metrics like root-mean-squared error and absolute error validated solution accuracy.
Conclusions:
- The novel numerical approach provides an effective and accurate solution for complex 3D elliptic PDEs.
- The method's ability to handle singular models and disperse errors enhances its utility in battery modeling.
- Fourth-order convergence on a quasi-variable grid signifies a significant advancement in numerical PDE analysis.
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