arXiv subjects

Thomas Christiansen

Publications and source records attributed to Thomas Christiansen.

5 recordsLinked to original sources

arXiv2026

Structure-preserving LDG methods for linear and nonlinear transport equations with gradient noise

We develop local discontinuous Galerkin (LDG) methods for conservation laws with heterogeneous stochastic fluxes, where the Stratonovich-driven transport terms may be linear or nonlinear. Such equations arise, for example, in simplified turbulence models, mean field games, and fluctuating hydrodynamics. Starting from the It\^{o} formulation, we construct semi-discretizations that build the cancellation mechanism of transport noise into the numerical method. At the discrete energy level, the second-order Stratonovich-It\^{o} correction is balanced by the quadratic variation, up to numerical flux terms, so that the hyperbolic stability structure is retained. Suitable numerical fluxes yield discrete energy conservation or energy dissipation, valid either pathwise or in expectation. The resulting high-order schemes are proved well posed through stability estimates combined with a Khasminskii-type argument, without imposing linear growth assumptions. Numerical experiments confirm stability and high-order accuracy.

math.NA↗

arXiv2025

Rate of convergence for numerical $α$-dissipative solutions of the Hunter-Saxton equation

We prove that $α$-dissipative solutions to the Cauchy problem of the Hunter-Saxton equation, where $α\in W^{1, \infty}(\mathbb{R}, [0, 1))$, can be computed numerically with order $\mathcal{O}(Δx^{{1}/{8}}+Δx^{β/{4}})$ in $L^{\infty}(\mathbb{R})$, provided there exist constants $C > 0$ and $β\in (0, 1]$ such that the initial spatial derivative $\bar{u}_{x}$ satisfies $\|\bar{u}_x(\cdot + h) - \bar{u}_x(\cdot)\|_2 \leq Ch^β$ for all $h \in (0, 2]$. The derived convergence rate is exemplified by a number of numerical experiments.

math.NA↗

arXiv2024

A Convergent Numerical Algorithm for $α$-Dissipative Solutions of the Hunter-Saxton Equation

A convergent numerical method for $α$-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 $L^{\infty}$ for all $t \geq 0$, while a subsequence of the energy density converges weakly for almost every time.

math.NA↗

arXiv2024

A numerical view on α-dissipative solutions of the Hunter-Saxton equation

We propose a new numerical method for $α$-dissipative solutions of the Hunter-Saxton equation, where $α$ belongs to $W^{1, \infty}(\mathbb{R}, [0, 1))$. 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 $C([0, T], L^{\infty}(\mathbb{R}))$ for any finite $T \geq 0$.

math.NA↗

arXiv2024

On the convergence rate of a numerical method for the Hunter-Saxton equation

We derive a robust error estimate for a recently proposed numerical method for $α$-dissipative solutions of the Hunter-Saxton equation, where $α\in [0, 1]$. In particular, if the following two conditions hold: i) there exist a constant $C > 0$ and $β\in (0, 1]$ such that the initial spatial derivative $\bar{u}_{x}$ satisfies $\|\bar{u}_x(\cdot + h) - \bar{u}_x(\cdot)\|_2 \leq Ch^β$ 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 $O(Δx^{\fracβ{8}})$ in $L^{\infty}(\mathbb{R})$. Moreover, if $α=0$, then the rate improves to $O(Δx^{\frac{1}{4}})$ without the above assumptions, and we also obtain a convergence rate for the associated energy measure - it converges with order $O(Δx^{\frac{1}{2}})$ in the bounded Lipschitz metric. These convergence rates are illustrated by several examples.

math.NA↗