-D = couple_discontinuously(D_legendre, uniform_mesh)</code></pre><p>Using <code>couple_discontinuously</code>, you can also construct upwind SBP operators by additionally passing <code>Val(:plus)</code> or <code>Val(:minus)</code>. Note that this construction results into non-uniformly distributed nodes and due to the discontinuous nature to repeated nodes at interfaces between elements.</p><h3 id="cg_sbp"><a class="docs-heading-anchor" href="#cg_sbp">Continuous Galerkin (CG) SBP</a><a id="cg_sbp-1"></a><a class="docs-heading-anchor-permalink" href="#cg_sbp" title="Permalink"></a></h3><p>Similarly to DG operators, CG operators can be constructed using <code>couple_continuously</code>. In contrast to the DG operators, <code>N</code> needs to be divisible by <code>p</code> and there are no repeated nodes at the interfaces.</p><h3 id="fourier_sbp"><a class="docs-heading-anchor" href="#fourier_sbp">Fourier/Spectral SBP</a><a id="fourier_sbp-1"></a><a class="docs-heading-anchor-permalink" href="#fourier_sbp" title="Permalink"></a></h3><p>Fourier or spectral SBP operators are constructed using Fourier basis functions. These operators can be used for problems with periodic boundary conditions. The key idea is to represent the solution in terms of its Fourier coefficients and to apply differentiation in the Fourier space. Note that these operators have dense derivative matrices and are therefore often more computationally expensive. In <a href="https://github.com/ranocha/SummationByPartsOperators.jl/">SummationByPartsOperators.jl</a>, they can be constructed with <code>fourier_derivative_matrix</code>.</p><h3 id="Variable-Coefficient-Operators"><a class="docs-heading-anchor" href="#Variable-Coefficient-Operators">Variable Coefficient Operators</a><a id="Variable-Coefficient-Operators-1"></a><a class="docs-heading-anchor-permalink" href="#Variable-Coefficient-Operators" title="Permalink"></a></h3><p>A special class of SBP operators is given by variable coefficient operators, which are discrete operators for the second derivative approximating terms of the form <span>$\partial_x(b \partial_x u)$</span>. Directly incorporating the variable coefficient <code>b</code> into the SBP operator is desirable compared to subsequent application of first-derivative operators <span>$D \textrm{diag}(\boldsymbol{b}) D$</span> because it leads to a more compact stencil and therefore improved numerical properties. You can use <code>var_coef_derivative_operator</code> with source <code>Mattsson2012</code> to construct such operators.</p><section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-LampertRanocha2025"><a class="tag is-link" href="#citeref-LampertRanocha2025">LampertRanocha2025</a>Lampert, Ranocha (2025): Structure-preserving numerical methods for two nonlinear systems of dispersive wave equations <a href="https://doi.org/10.1007/s44207-025-00006-3">DOI: 10.1007/s44207-025-00006-3</a></li><li class="footnote" id="footnote-SvärdNordström2019"><a class="tag is-link" href="#citeref-SvärdNordström2019">SvärdNordström2019</a>Svärd, Nordström (2019): On the convergence rates of energy-stable finite-difference schemes <a href="https://doi.org/10.1016/j.jcp.2019.07.018">DOI: 10.1016/j.jcp.2019.07.018</a></li><li class="footnote" id="footnote-SvärdNordström2021"><a class="tag is-link" href="#citeref-SvärdNordström2021">SvärdNordström2021</a>Svärd, Nordström (2021): Convergence of energy stable finite-difference schemes with interfaces <a href="https://doi.org/10.1016/j.jcp.2020.110020">DOI: 10.1016/j.jcp.2020.110020</a></li><li class="footnote" id="footnote-Gassner2013"><a class="tag is-link" href="#citeref-Gassner2013">Gassner2013</a>Gassner (2013): A Skew-Symmetric Discontinuous Galerkin Spectral Element Discretization and Its Relation to SBP-SAT Finite Difference Methods <a href="https://epubs.siam.org/doi/10.1137/120890144">DOI: 10.1137/120890144</a></li><li class="footnote" id="footnote-RanochaMitsotakisKetcheson2021"><a class="tag is-link" href="#citeref-RanochaMitsotakisKetcheson2021">RanochaMitsotakisKetcheson2021</a>Ranocha, Mitsokatis, Ketcheson (2021): A broad class of conservative numerical methods for dispersive wave equations <a href="https://doi.org/10.4208/cicp.oa-2020-0119">DOI: 10.4208/cicp.oa-2020-0119</a></li></ul></section></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../basic_example/">« Basic Example</a><a class="docs-footer-nextpage" href="../dispersion/">Dispersion »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.17.0 on <span class="colophon-date" title="Wednesday 8 April 2026 16:11">Wednesday 8 April 2026</span>. 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