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This book explains how numerical methods turn mathematical models of physical processes into calculations that can be performed on a computer. The authors begin with fundamental properties of partial differential equations, including questions of existence, uniqueness and conservation, before examining how a continuous problem can be replaced by a finite system of approximations.
The central chapters develop finite difference, finite volume and finite element approaches. Worked mathematical arguments connect the construction of a numerical scheme with questions about consistency, accuracy and the behaviour of its solutions. Additional discussions address the linear and nonlinear algebraic systems that arise after discretisation and the treatment of time-dependent problems.
Written for engineers and scientists, the text keeps the relationship between a physical model and its numerical implementation in view. Chapter objectives, examples, summaries and exercises support independent study. The work was first printed in 2019; this complete publisher-issued open edition preserves the later explanatory material, original equations and diagrams, and the contributions of all four credited authors.