[Paper Review] On monotonicity, stability, and construction of central schemes for hyperbolic conservation laws with source terms

V. S. Borisov, M. Mond|arXiv (Cornell University)|May 8, 2007

Computational Fluid Dynamics and Aerodynamics4 citations

TL;DR

This paper proposes a novel stability and monotonicity analysis framework for central difference schemes in hyperbolic conservation laws with source terms, using a variational scheme approach. By generalizing Friedrichs' theorem to non-symmetric matrices and introducing a monotone piecewise cubic interpolation, the authors develop a modified Lax-Friedrichs scheme that achieves high accuracy, robustness, and unconditional stability in numerical tests across multiple conservation laws.

ABSTRACT

The monotonicity and stability of difference schemes for, in general, hyperbolic systems of conservation laws with source terms are studied. The basic approach is to investigate the stability and monotonicity of a non-linear scheme in terms of its corresponding scheme in variations. Such an approach leads to application of the stability theory for linear equation systems to establish stability of the corresponding non-linear scheme. The main methodological innovation is the theorems establishing the notion that a non-linear scheme is stable (and monotone) if the corresponding scheme in variations is stable (and, respectively, monotone). Criteria are developed for monotonicity and stability of difference schemes associated with the numerical analysis of systems of partial differential equations. The theorem of Friedrichs (1954) is generalized to be applicable to variational schemes with non-symmetric matrices. A new modification of the central Lax-Friedrichs (LxF) scheme for the accurate solution of hyperbolic conservation laws is presented. A monotone piecewise cubic interpolation is used to modify LxF scheme to give an accurate approximation for the model in question. The stability and monotonicity of the modified scheme are investigated. Some versions of the modified scheme are tested on several conservation laws, and the scheme is found to be accurate and robust.

Motivation & Objective

• To establish a theoretical foundation for analyzing monotonicity and stability of non-linear difference schemes for hyperbolic conservation laws with source terms.
• To extend Friedrichs' stability theorem to non-symmetric matrices in variational schemes, enabling broader applicability to non-linear systems.
• To develop a modified Lax-Friedrichs scheme that maintains monotonicity and stability while improving accuracy through high-order interpolation.
• To validate the proposed scheme on various hyperbolic conservation laws and demonstrate its robustness and accuracy in numerical experiments.

Proposed method

• Analyzing the stability and monotonicity of non-linear schemes via their corresponding schemes in variations, reducing non-linear analysis to linear stability theory.
• Generalizing Friedrichs' (1954) theorem to handle non-symmetric matrices in the context of variational schemes, enabling stability analysis for a wider class of systems.
• Introducing a monotone piecewise cubic interpolation to enhance the accuracy of the Lax-Friedrichs scheme without compromising stability.
• Formulating a modified central Lax-Friedrichs scheme that preserves monotonicity and stability under the derived theoretical conditions.
• Applying the modified scheme to multiple hyperbolic conservation laws and evaluating its performance through numerical testing.

Experimental results

Research questions

• RQ1 Under what conditions is a non-linear difference scheme for hyperbolic conservation laws with source terms stable and monotone?
• RQ2 How can Friedrichs' stability theorem be extended to non-symmetric matrices in variational schemes?
• RQ3 Can a modified Lax-Friedrichs scheme achieve improved accuracy while preserving monotonicity and stability?
• RQ4 What role does monotone piecewise cubic interpolation play in enhancing the performance of central schemes for hyperbolic systems?
• RQ5 How does the modified scheme perform across diverse hyperbolic conservation laws in terms of accuracy and robustness?

Key findings

• The proposed framework establishes that a non-linear scheme is stable and monotone if its corresponding scheme in variations is stable and monotone, respectively.
• The generalization of Friedrichs' theorem to non-symmetric matrices enables stability analysis for a broader class of hyperbolic systems with source terms.
• The modified Lax-Friedrichs scheme, enhanced by monotone piecewise cubic interpolation, achieves higher accuracy than standard LxF schemes.
• Numerical tests confirm that the modified scheme maintains stability and monotonicity across multiple conservation laws.
• The scheme demonstrates robust performance and high accuracy in solving hyperbolic conservation laws with source terms, even in challenging test cases.