A Numerical Method for the Porous Medium Equation Using the Onsager Variational Principle and Moving Mesh Approach

Concetti Chiave

The authors derive a novel numerical method for the porous medium equation by leveraging the Onsager variational principle and a moving mesh finite element approach. The method maintains the energy dissipation structure and accurately captures the waiting time phenomenon.

Sintesi

The paper introduces a new approach to solving the porous medium equation (PME) using a moving mesh finite element method that is based on the Onsager variational principle.
Key highlights:

The continuous and discrete problems are formulated based on the Onsager principle, which maintains the energy dissipation structure in the semi-discrete and fully implicit discrete schemes.
An explicit fully decoupled scheme is developed where only a few linear equations are solved sequentially in each time step.
The numerical schemes exhibit optimal convergence rates when the initial mesh is appropriately selected to ensure accurate approximation of the initial data.
The method naturally captures the waiting time phenomena without requiring any manual intervention.
Two semi-discrete schemes are derived - one that maintains the mass conservation property and one that is more efficient but does not strictly conserve mass.
Numerical examples in 1D and 2D demonstrate the effectiveness of the proposed methods.

Statistiche

The paper does not provide specific numerical data or metrics to support the key claims. However, it does state that the proposed methods exhibit optimal convergence rates and accurately capture the waiting time phenomenon.

Citazioni

None.

Approfondimenti chiave tratti da

A Moving Mesh Method for Porous Medium Equation by the Onsager Variational Principle

by Si Xiao,Xian... alle arxiv.org 04-01-2024

https://arxiv.org/pdf/2403.20030.pdf

Domande più approfondite

How sensitive are the proposed numerical schemes to the choice of the initial mesh

The sensitivity of the proposed numerical schemes to the choice of the initial mesh is a crucial aspect to consider for achieving optimal performance. The initial mesh plays a significant role in determining the accuracy and convergence of the numerical solution. Guidelines for selecting an appropriate initial mesh can help mitigate potential issues and enhance the efficiency of the method.
To ensure optimal performance:

Density of Nodes: The initial mesh should have a sufficient density of nodes, especially in regions where the solution exhibits rapid changes or gradients. A finer mesh can capture details more accurately.
Boundary Conditions: The mesh should be refined near boundaries or regions of interest to accurately capture boundary conditions and the behavior of the solution in those areas.
Adaptive Mesh Refinement: Consider using adaptive mesh refinement techniques that dynamically adjust the mesh based on the solution's behavior during the simulation. This can help concentrate computational resources where they are most needed.
Regular Grid Structure: Maintaining a regular grid structure can aid in simplifying the implementation and analysis of the numerical method.

By following these guidelines and possibly incorporating adaptive mesh refinement strategies, the choice of the initial mesh can be optimized to enhance the performance and accuracy of the numerical schemes.

Can guidelines be provided for selecting an appropriate initial mesh to ensure optimal performance

While the Onsager variational principle-based approach offers several advantages for solving the porous medium equation, there are potential drawbacks and limitations compared to other numerical methods:

Complexity: The implementation of the Onsager variational principle-based approach may involve intricate mathematical derivations and computational procedures, making it more complex than some traditional numerical methods.

Computational Cost: The implicit schemes derived from the Onsager principle may require solving nonlinear systems of equations, leading to increased computational cost and complexity compared to explicit schemes.

Convergence: The convergence properties of the numerical schemes based on the Onsager variational principle may vary depending on the problem characteristics and discretization choices, potentially requiring careful analysis and tuning.

Generalizability: The applicability of the Onsager variational principle-based approach to a broader class of nonlinear partial differential equations beyond the porous medium equation may be limited, as the method's effectiveness could be specific to the characteristics of the PME.

While the Onsager variational principle provides a unique and insightful framework for tackling the PME, researchers should be mindful of these limitations when considering its use in practical applications.

What are the potential drawbacks or limitations of the Onsager variational principle-based approach compared to other numerical methods for the porous medium equation

The proposed framework based on the Onsager variational principle can be extended to solve other types of nonlinear partial differential equations that exhibit similar challenges, such as free boundaries and singularities. By leveraging the fundamental principles of the Onsager variational approach, researchers can develop numerical schemes for a wide range of nonlinear PDEs with complex dynamics.
Extensions to other equations could involve adapting the variational formulation to suit the specific characteristics of the new equation. This may include modifying the dissipation function, energy functional, and constraints to align with the properties of the target equation. Additionally, the discretization and solution strategies can be tailored to address the unique challenges posed by different types of nonlinear PDEs.
By applying the principles and methodologies inspired by the Onsager variational principle, researchers can explore innovative numerical approaches for a diverse set of nonlinear PDEs, paving the way for advancements in computational modeling and simulation across various scientific and engineering domains.