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High-order finite difference methods

In: An Introduction to Scientific Computing

Author

Listed:
  • 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)

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.

Suggested Citation

  • Ionut Danaila & Pascal Joly & Sidi Mahmoud Kaber & Marie Postel, 2023. "High-order finite difference methods," Springer Books, in: An Introduction to Scientific Computing, edition 2, chapter 0, pages 145-178, Springer.
    • Handle: RePEc:spr:sprchp:978-3-031-35032-0_7
    DOI: 10.1007/978-3-031-35032-0_7
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