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Adapted Lie Splitting Method for Convection-Diffusion Problems with Singular Convective Term


Core Concepts
The adapted Lie splitting method overcomes the instability issues encountered in the classical Lie splitting approach when applied to convection-diffusion problems with unbounded convective terms.
Abstract
The paper proposes a new splitting method, called the "Adapted Lie Splitting Method", to tackle the instability issues observed in the standard Lie splitting approach when applied to convection-diffusion problems with unbounded convective terms. Key highlights: The authors introduce an adapted splitting approach that decomposes the unbounded convective coefficient into a controlled component and a bounded component. This reformulation effectively avoids the inherent instability conditions encountered. Theoretical convergence analysis is provided, proving that the adapted Lie splitting method is a first-order convergent method under suitable assumptions on the convective term. Numerical experiments in both two and three dimensions demonstrate the effectiveness of the proposed splitting method, showing significant improvements in stability and accuracy compared to the classical Lie splitting approach. The adapted Lie splitting method offers a promising technique for achieving numerical stability and enhancing accuracy in broader scenarios characterized by unbounded convective terms.
Stats
The convective coefficient c(x) belongs to the weak-Marcinkiewicz space LN,∞(Ω). The adapted Lie splitting method achieves an accuracy factor of approximately 2 smaller than the classical Lie splitting method for time-independent boundary conditions, and a factor of 4 for time-dependent boundary conditions. In the case of homogeneous boundary conditions, the adapted Lie splitting method shows an accuracy improvement of approximately a factor of 18 over the classical Lie splitting method.
Quotes
"Splitting methods are a widely used numerical scheme for solving convection-diffusion problems." "However, they may lose stability in some situations, particularly when applied to convection-diffusion problems in the presence of an unbounded convective term." "The adapted Lie splitting method offers a promising technique for achieving numerical stability and greatly enhancing accuracy in broader scenarios characterized by unbounded convective terms."

Deeper Inquiries

How can the proposed adapted Lie splitting method be extended to higher-order splitting schemes, such as Strang splitting, to further improve the convergence rate

The proposed adapted Lie splitting method can be extended to higher-order splitting schemes, such as Strang splitting, by incorporating additional correction terms to account for the unbounded convective coefficients. In the context of Strang splitting, the method involves splitting the evolution operator into smaller operators and applying them sequentially. To improve the convergence rate, one can introduce higher-order correction terms in the splitting scheme to account for the unbounded convective terms more accurately. By incorporating higher-order corrections, the method can achieve a higher level of accuracy and convergence, leading to more efficient numerical simulations.

What are the potential challenges and limitations in applying the adapted Lie splitting approach to more complex convection-diffusion systems, such as those involving nonlinear or time-dependent diffusion operators

When applying the adapted Lie splitting approach to more complex convection-diffusion systems involving nonlinear or time-dependent diffusion operators, several challenges and limitations may arise. One potential challenge is the increased computational complexity due to the nonlinear nature of the diffusion operators, which may require more sophisticated numerical techniques to handle efficiently. Additionally, the presence of time-dependent diffusion operators can introduce additional instabilities and numerical difficulties, requiring careful consideration and adjustment of the splitting method to ensure stability and accuracy. Furthermore, the adaptation of the method to handle nonlinear diffusion terms may require the development of specialized algorithms and strategies to address the complexities introduced by the nonlinearity.

Can the ideas behind the adapted Lie splitting method be applied to other types of partial differential equations beyond convection-diffusion problems, where the presence of unbounded coefficients poses numerical challenges

The ideas behind the adapted Lie splitting method can be applied to other types of partial differential equations beyond convection-diffusion problems where the presence of unbounded coefficients poses numerical challenges. For example, the method can be extended to reaction-diffusion equations, where the reaction terms may exhibit similar challenges as the unbounded convective terms in convection-diffusion problems. By adapting the splitting approach to handle unbounded coefficients in reaction-diffusion systems, one can improve the stability and accuracy of numerical simulations in these contexts. Additionally, the method can be applied to other types of PDEs with unbounded coefficients, such as transport equations or wave equations, to address numerical challenges associated with unboundedness and improve the efficiency of numerical computations.
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