Thomas Christiansen

• Christiansen, Thomas (2026). On the convergence rate of a numerical method for the Hunter–Saxton equation. IMA Journal of Numerical Analysis. ISSN 0272-4979. doi: 10.1093/imanum/draf119. Full text in Research Archive Show summary

  We derive a robust error estimate for a recently proposed numerical method for $\alpha $-dissipative solutions of the Hunter–Saxton equation, where $\alpha \in [0, 1]$. In particular, if the following two conditions hold: (i) there exist a constant $C> 0$ and $\beta \in (0, 1]$ such that the initial spatial derivative $\bar{u}_{x}$ satisfies $\|\bar{u}_{x}(\cdot + h) - \bar{u}_{x}(\cdot )\|_{2} \leq Ch^{\beta }$ for all $h \in (0, 2]$, and (ii) the singular continuous part of the initial energy measure is zero then the numerical wave profile converges with order $\mathscr{O}({\varDelta x}^{{\beta }/{8}})$ in $L^{\infty }(\mathbb{R})$. Moreover, if $\alpha =0$ then the rate improves to $\mathscr{O}({\varDelta x}^{{1}/{4}})$ without the above assumptions, and we also obtain a convergence rate for the associated energy measure—it converges with order $\mathscr{O}({\varDelta x}^{{1}/{2}})$ in the bounded Lipschitz metric. These convergence rates are illustrated by several examples.

• Christiansen, Thomas & Grunert, Katrin (2025). Rate of convergence for numerical α-dissipative solutions of the Hunter–Saxton equation. Numerische Mathematik. ISSN 0029-599X. 157(5), p. 1643–1694. doi: 10.1007/s00211-025-01482-7. Full text in Research Archive
• Grunert, Katrin & Christiansen, Thomas (2025). A numerical view on α -dissipative solutions of the Hunter- Saxton equation. ESAIM: Mathematical Modelling and Numerical Analysis (ESAIM: M2AN). ISSN 2822-7840. 59(1), p. 579–612. doi: 10.1051/m2an/2024083. Full text in Research Archive Show summary

  We propose a new numerical method for α-dissipative solutions of the Hunter–Saxton equation, where α belongs to W1,∞(R, [0, 1)). The method combines a projection operator with a generalized method of characteristics and an iteration scheme, which is based on enforcing minimal time steps whenever breaking times cluster. Numerical examples illustrate that these minimal time steps increase the efficiency of the algorithm substantially. Moreover, convergence of the wave profile is shown in C([0, T], L∞(R)) for any finite T ≥ 0.

• Christiansen, Thomas; Grunert, Katrin; Nordli, Anders Samuelsen & Solem, Susanne (2024). A Convergent Numerical Algorithm for α-Dissipative Solutions of the Hunter–Saxton Equation. Journal of Scientific Computing. ISSN 0885-7474. 99(1). doi: 10.1007/s10915-024-02479-4. Full text in Research Archive Show summary

  A convergent numerical method for -dissipative solutions of the Hunter–Saxton equation is derived. The method is based on applying a tailor-made projection operator to the initial data, and then solving exactly using the generalized method of characteristics. The projection step is the only step that introduces any approximation error. It is therefore crucial that its design ensures not only a good approximation of the initial data, but also that errors due to the energy dissipation at later times remain small. Furthermore, it is shown that the main quantity of interest, the wave profile, converges in for all , while a subsequence of the energy density converges weakly for almost every time.

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• Christiansen, Thomas (2024). A convergent numerical method for the Hunter–Saxton equation. Full text in Research Archive
• Christiansen, Thomas (2024). A convergent numerical method for α-dissipative solutions of the Hunter–Saxton equation. Full text in Research Archive
• Christiansen, Thomas (2024). Numerical α-dissipative solutions of the Hunter-Saxton equation and their rate of convergence. Full text in Research Archive
• Christiansen, Thomas (2023). A numerical algorithm for α-dissipative solutions of the Hunter-Saxton equation. Full text in Research Archive
• Christiansen, Thomas (2023). A numerical algorithm for 𝛼-dissipative solutions of the Hunter–Saxton equation. Full text in Research Archive
• Christiansen, Thomas (2023). A convergent numerical method for the Hunter-Saxton equation. Full text in Research Archive
• Christiansen, Thomas (2022). A numerical algorithm for traveling wave solutions of the Hunter-Saxton equation. Full text in Research Archive

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