Constructing a New Class of Backward Difference Formulas and Implicit-Explicit Schemes for Parabolic Equations with Improved Stability

Concetti Chiave

A new class of BDF and IMEX schemes is constructed based on Taylor expansions at time tn+β, with β > 1 being a tunable parameter. These new schemes allow for larger time steps at higher-order accuracy compared to classical schemes, particularly for stiff problems.

Sintesi

The authors construct a new class of backward difference formula (BDF) and implicit-explicit (IMEX) schemes for solving parabolic type equations. The key idea is to base the schemes on Taylor expansions at time tn+β, where β > 1 is a tunable parameter, rather than the classical case of β = 1.

The main highlights and insights are:

1. The new schemes generalize the classical BDF and IMEX schemes, with the same computational effort but improved stability properties. By choosing a suitable β > 1, the stability regions of the higher-order schemes can be significantly enlarged compared to the classical case.

2. The authors identify an explicit and uniform multiplier for the new class of BDF and IMEX schemes up to fourth-order, which is crucial for establishing rigorous stability and error analysis using energy arguments.

3. For linear parabolic equations, the new schemes are shown to be unconditionally stable. For nonlinear parabolic equations, the stability and error analysis demonstrate that the new schemes become less restrictive as β increases, especially compared to the classical case of β = 1.

4. Numerical examples are provided to validate the theoretical findings, showing the advantages of the new schemes in terms of allowing larger time steps at higher-order accuracy.

Statistiche

The authors provide the explicit coefficients for the new BDF and IMEX schemes up to fourth-order in equations (2.12), (2.13) and (2.14).

Citazioni

"When applying the classical multistep schemes for solving differential equations, one often faces the dilemma that smaller time steps are needed with higher-order schemes, making it impractical to use high-order schemes for stiff problems."
"We construct in this paper a new class of BDF and implicit-explicit (IMEX) schemes for parabolic type equations based on the Taylor expansions at time tn+β with β > 1 being a tunable parameter."
"These new schemes, with a suitable β, allow larger time steps at higher-order for stiff problems than that is allowed with a usual higher-order scheme."

Approfondimenti chiave tratti da

On a new class of BDF and IMEX schemes for parabolic type equations

by Fukeng Huang... alle arxiv.org 05-02-2024

https://arxiv.org/pdf/2405.00300.pdf

Domande più approfondite

How can the ideas behind the new class of BDF and IMEX schemes be extended to other types of numerical methods beyond multistep schemes

The ideas behind the new class of BDF and IMEX schemes can be extended to other types of numerical methods beyond multistep schemes by considering different types of differential equations and their corresponding discretization methods. For example, the concept of using a tunable parameter in the Taylor expansions to improve stability and allow for larger time steps can be applied to other time integration schemes, such as Runge-Kutta methods or exponential integrators. By adjusting the parameter in the numerical schemes based on the specific characteristics of the differential equations, similar benefits in terms of stability and efficiency can be achieved in various numerical methods.

What are the potential limitations or drawbacks of the new schemes, and under what circumstances might the classical schemes still be preferable

The new schemes introduced in the context may have potential limitations or drawbacks in certain scenarios. One limitation could be the increased computational complexity associated with the higher-order schemes, especially when dealing with large-scale systems or time-dependent problems. Additionally, the need to determine the optimal value of the tunable parameter β for different types of equations and problem settings could pose a challenge in practical implementations. In cases where the stiffness of the problem is not a significant concern or when computational resources are limited, the classical schemes may still be preferable due to their simplicity and ease of implementation. Moreover, for problems where the stiffness is not pronounced, using higher-order schemes with larger time steps may not provide significant advantages over lower-order schemes.

Can the techniques used to derive the explicit multipliers and telescoping formulae be applied to develop similar results for other classes of differential equations beyond parabolic type

The techniques used to derive the explicit multipliers and telescoping formulae for the new class of BDF and IMEX schemes can be applied to develop similar results for other classes of differential equations beyond parabolic type. By analyzing the stability and error properties of different numerical methods using energy arguments and explicit multipliers, it is possible to extend these analyses to various types of differential equations, including hyperbolic, elliptic, and mixed-type equations. The key lies in adapting the energy stability analysis and multiplier construction to the specific characteristics and properties of the differential equations under consideration. This approach can provide valuable insights into the stability and convergence properties of numerical methods for a wide range of differential equation problems.