High-order finite difference methods

Ionut Danaila (), Pascal Joly, Sidi Mahmoud Kaber and Marie Postel
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Ionut Danaila: Université de Rouen Normandie, CNRS, Laboratoire de mathématiques Raphaël Salem
Pascal Joly: Laboratoire Jacques-Louis Lions
Sidi Mahmoud Kaber: Sorbonne Université, CNRS, Université Paris Cité, Laboratoire Jacques-Louis Lions
Marie Postel: Sorbonne Université, CNRS, Université Paris Cité, Laboratoire Jacques-Louis Lions

Chapter Chapter 7 in An Introduction to Scientific Computing, 2023, pp 145-178 from Springer

Abstract: Abstract Finite difference (FD) methods are very popular for solving partial differential equations (PDEs) because of their simplicity. A simple but powerful mathematical tool, namely the Taylor series expansion, is necessary to derive FD schemes to approximate derivatives. We present in this chapter a general method to derive high-order FD schemes with arbitrary approximation. In addition to classical explicit schemes (the approximation is calculated explicitly using known values), we also introduce implicit schemes for which the approximation of derivatives involves the resolution of a linear system. The very popular FD compact schemes, with spectral-like precision, are also explained in detail. As an application, we compare explicit/implicit high-order FD schemes for solving the 1D linear heat equation with Dirichlet or periodic boundary conditions.

Date: 2023
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DOI: 10.1007/978-3-031-35032-0_7

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