The Solution of Partial Differential Equations by Finite Difference Approximations: Analysing the Relative Performance of Differing Numerical Finite Difference Schemes using Taylor Series Expansions

A comprehensive performance analysis of the Finite Difference Method for the solution of Partial Differential Equations. Providing an in-depth understanding of; Finite Difference Methods, their applications, theoretical basis, the full derivation of Taylor Series Expansions and the construction of a working Computational Domain Grid System. Furthermore, detailing and showing how to effectively employ the Finite Difference Method, through the implementation of Finite Difference Schemes, to obtain accurate, stable and consistent numerical solutions for Partial Differential Equations, which model a multitude of varying dynamic processes. Moreover, it contains a detailed, thorough performance analysis investigation of three different Finite Difference Method schemes, when they are employed to obtain accurate numerical solutions for a fluid flow heat transfer process that is modelled by a first order Partial Differential Equation. These three schemes are the Forward-Time-Backwards-Space, Lax and Lax Wendroff Finite Difference Method schemes. Additionally, it explains the criteria that is required for optimal scheme stability, consistency and convergence. A brief breakdown of what the book contains;•A Description of the processes required to conduct an effective performance analysis of Finite Difference Method Schemes. •It specifies and explains the Forward-Time-Backwards-Space, Lax and Lax-Wendroff Finite Difference Scheme equations.•Explanations of the concepts of Finite Difference Method Stability, Consistency and Convergence. •The full derivations of the Taylor Series Expansions of the Forward-Time-Backwards-Space, Lax and Lax-Wendroff Finite Difference Scheme equations.•The development of an effective Finite Difference Method Computational Grid System, that can be used to calculate accurate numerical solutions for Partial Differential Equations. •A comprehensive end-to-end performance analysis of the three schemes for a fluid flow heat transfer process.•A discussion of the usefulness of the Finite Difference Method for solving Partial Differential Equations.•An overview of how to select an optimal Finite Difference Method scheme for accurate numerical solutions.You will gain valuable knowledge of the Finite Difference Method and its applications, expanding your expertise and intellect in this area of mathematics. Additionally, it will enable you to develop a systematic understanding of how to use Finite Difference Schemes to solve Partial Differential Equations and obtain accurate numerical solutions for dynamic processes. The book is self-contained allowing you to understand and conduct a Finite Difference Method performance analysis, so that you can apply the concepts to any process that is modelled by hyperbolic Partial Differential Equations. Furthermore, it is particularly valuable to; academics, educators, scholars, engineering industry professionals, and students. Especially, postgraduate Master’s and undergraduate students. Assisting those who work/operate/study in the fields of Aerodynamics, Mathematics, Aerospace, Fluid Dynamics and Fluid Mechanics. Overall, this book will save you countless hours of research and reading, since the information contained within is distilled, concentrated and assimilated in an effective manner to help you to develop a deep understanding regarding the performance of the Finite Difference Method.