InternalModesFiniteDifference

Solve the vertical eigenvalue problems with finite-difference matrices.

---

Declaration

classdef InternalModesFiniteDifference < InternalModesBase

Overview

InternalModesFiniteDifference discretizes the fixed-\(K\) and fixed-\(\omega\) eigenvalue problems directly on the supplied depth grid. It uses arbitrary-order finite-difference stencils generated from the Fornberg weights algorithm and optionally interpolates the resulting modes onto a separate output grid.

This class is mainly useful as a baseline against the spectral methods discussed in Section 3 of Early, Lelong, and Smith (2020), where the manuscript explains why finite differencing becomes less attractive when many frequencies or wavenumbers are required.

im = InternalModesFiniteDifference(rho=rho, zIn=zIn, zOut=zOut, latitude=latitude, orderOfAccuracy=4);
[F, G, h, omega] = im.modesAtWavenumber(2*pi/1000);

Topics

• Create and initialize modes
  • InternalModesFiniteDifference Initialize the finite-difference solver.
  • orderOfAccuracy Formal order of accuracy for the finite-difference stencils.
• Compute modes
  • bottomModesAtWavenumber Return the bottom SQG mode at fixed horizontal wavenumber.
  • boundaryModesAtWavenumber Return either the surface or bottom SQG mode at fixed horizontal wavenumber.
  • surfaceModesAtWavenumber Return the surface SQG mode at fixed horizontal wavenumber.

Developer Topics

These items document internal implementation details and are not part of the primary public API.

• Developer topics
  • ApplyBoundaryConditions Apply the current lower and upper boundary conditions to an EVP pair.
  • Diff1 First-derivative finite-difference matrix.
  • Diff2 Second-derivative finite-difference matrix.
  • FiniteDifferenceMatrix Build a finite-difference differentiation matrix with boundary stencils.
  • InitializeOutputTransformation Prepare the interpolation from the differentiation grid to the public output grid.
  • ModesFromGEP Solve a generalized EVP and map its modes onto the public grid.
  • N2_z_diff Buoyancy frequency squared sampled on z_diff.
  • NormalizeModes Normalize finite-difference modes using the active convention.
  • T_out Transformation from z_diff functions to the public output grid.
  • n Number of differentiation-grid points.
  • rho_z_diff Background density sampled on z_diff.
  • weights Return Fornberg finite-difference weights for one stencil location.
  • z_diff Depth grid used for differentiation and the generalized EVP.

---