Abstract
In this chapter, we mainly discuss the topic of the compatibility, convergence, and stability of difference schemes for linear initial value problems. Firstly, some theoretical bases of functional analysis such as the definition and properties of linear space, linear operator, and norm are introduced. In the stability analysis part, we focus on the Von-Neumann stability analysis and introduce other methods, including maximum modulus method, discrete perturbation method, energy method, and Hirt heuristic method, to analyse the numerical stability. Besides, a simple method using difference operator transform to calculate the transition factor is proposed, and an adequate discussion about various forms of stability conditions and the Lax equivalence theorem is given.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Konangi S, Palakurthi et al (2018) Von Neumann stability analysis of first-order accurate discretization schemes for one-dimensional (1D) and two-dimensional (2D) fluid flow equations. Comput Math Appl Int J 75(2):643–665
Konangi S, Palakurthi NK , Ghia U (2016) Von neumann stability analysis of a segregated pressure-based solution scheme for one-dimensional and two-dimensional flow equations. J Fluids Eng 138(10):101401
Chen C (1990) A new method of finding an amplification factor by the Fourier transform of a difference operator. J Huazhong Univ Sci Technol 018(005):13–20
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2024 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.
About this chapter
Cite this chapter
Hou, G., Chen, C., Qin, S., Gao, Y., Wang, K. (2024). The Compatibility, Convergence, and Stability of Difference Schemes. In: Computational Fluid Dynamics. Engineering Applications of Computational Methods, vol 20. Springer, Singapore. https://doi.org/10.1007/978-981-97-0349-4_2
Download citation
DOI: https://doi.org/10.1007/978-981-97-0349-4_2
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-97-0348-7
Online ISBN: 978-981-97-0349-4
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)