A numerical tour of wave propagation |

To approximate the 1st-order derivatives as accurate as possible, we express it in the following

(32) |

where . Substituting the and with (29) for results in

(33) |

Thus, taking first terms means

(34) |

In matrix form,

(35) |

The above matrix equation is Vandermonde-like system: , . The Vandermonde matrix

(36) |

in which , has analytic solutions. can be solved using the specific algorithms, see Bjorck (1996). And we obtain

(37) |

The MATLAB code for solving the 2

In general, the stability of staggered-grid difference requires that

(38) |

Define . Then, we have

In the 2nd-order case, numerical dispersion is limited when

(39) |

The 4th-order dispersion relation is:

(40) |

A numerical tour of wave propagation |

2021-08-31