Applying the method of characteristics (MOC) to a hyperbolic second order partial differential equation (PDE) in two variables yields two systems of ordinary differential equations (ODEs) describing the evolution of the solution along two families of characteristics. In general the ODE systems are mutually coupled, implying a characteristic depends on both the other characteristic and on the solution of the PDE. We applied the MOC to the 2D hyperbolic Monge-Ampère equation using the $x$-coordinate as parametrization. The solution evolves along the characteristics and henceforth the number, type and location of the necessary boundary conditions (implicitly) depends on the characteristics over the entire domain. A criterion for the boundary conditions is formulated. Based on the coupled systems of ODEs we formulate a numerical solution method. We employ explicit one-step methods (Euler, Runge-Kutta) to integrate the ODEs systems. Subsequently, we apply B-spline interpolation along vertical grid lines to approximate the solution in grid points. Using an adaptive integration step we bound the vertical distance between (approximated) adjacent characteristics and, hence, the errors of the interpolation. Subsequently, the order of accuracy of the numerical scheme is known and the stability of the scheme is obtained. The numerical scheme attains the theoretical order of convergence and converges to computer precision for multiple examples if the numerical grid is fine enough. Moreover, the numerical method is able to handle complicated cases with varying numbers of boundary conditions. Reformulation of the PDE as an integral equation yields a proxy for the residual which attains the theoretical convergence rates.

We consider semilinear hyperbolic systems $$ \partial_t \textbf{u} + A(\partial) \textbf{u} + \frac{1}{\varepsilon} E \textbf{u} = \varepsilon T(\textbf{u},\textbf{u},\textbf{u}), \quad t \in \left[0,\frac{t_{\text{end}}}{\varepsilon}\right], \quad x \in \mathbb{R}^d $$ with a small parameter $0 < \varepsilon \ll 1$ and highly oscillatory initial data $$ \textbf{u}(0,x) = 2 p(x) \cos\left(\tfrac{\kappa \cdot x}{\varepsilon}\right). %p(x) \exp\left(\text{i} \frac{\kappa \cdot x}{\varepsilon} \right) + p(x) \exp\left(-\text{i} \frac{\kappa \cdot x}{\varepsilon} \right). $$ $T(\cdot,\cdot,\cdot)$ is a trilinear nonlinearity, $E\in\mathbb{R}^{s \times s} $ is a skew-symmetric matrix, and $A(\partial)$ denotes the differential operator $ A(\partial) = \sum_{\mu = 1}^{d} A_\mu \partial_\mu $ with symmetric matrices $A_1, \ldots, A_d \in \mathbb{R}^{s \times s}$ ($d,s \in \mathbb{N}$). As initial data we consider wavetrains, which involve a wave vector $\kappa \in \mathbb{R}^{d} \setminus \{0\}$ and a smooth envelope function $p \, : \, \mathbb{R}^{d} \rightarrow \mathbb{R}^{s}$.

Physically relevant solutions oscillate rapidly in time and space due to the small parameter $\varepsilon$ which is contained in the PDE and in the initial data. Moreover, solutions have to be computed on a long time interval $[0, t_{\text{end}} / \varepsilon]$ for some $t_{\text{end}} >0$. As a consequence, solving this PDE with initial data of the given form is a challenging task.

A popular approach in the literature is the slowly varying envelope approximation which provides a possibility to approximate the solution $\textbf{u}$ with an accuracy of $\mathcal{O}(\varepsilon)$. The envelope equation avoids oscillations in space which makes this approximation attractive for numerical computations. Under slightly stronger assumptions we prove that the accuracy of the slowly varying envelope approximation is even $\mathcal{O}(\varepsilon^2)$. This is a significant improvement.

We consider the Landau-Lifshitz-Gilbert-equation (LLG) on a bounded domain $\Omega$ coupled to the Maxwell equations on the whole 3D space.

We propose a numerical algorithm based on convolution quadrature and show a-priori error bounds in the situation of a sufficiently regular exact solution. To that end, we combine the convergence results for the LLG equation via linearly implicit backward difference formulae and the leapfrog and convolution quadrature coupling for the Maxwell system. The precise method of coupling allows us to reduce the full nonlinear system to a series of linear solves and still achieves the same convergence rates as in the uncoupled cases.

Numerical experiments via FEniCS and Bempp illustrate the theoretical results and demonstrate the applicability of the method. Furthermore we present an adaptive algorithm and numerical experiments to adaptively approximate the LLG equation in space and time.

The Onsager framework for linear irreversible thermodynamics provides a thermodynamically consistent model of mass transport in a phase consisting of multiple species, via the Stefan–Maxwell equations, but a complete description of the overall transport problem necessitates also solving the momentum equations for the flow velocity of the medium. We derive a novel nonlinear variational formulation of this coupling, called the (Navier–)Stokes–Onsager–Stefan–Maxwell system, which governs molecular diffusion and convection within a non-ideal, single-phase fluid composed of multiple species, in the regime of low Reynolds number in the steady state. We propose an appropriate Picard linearization posed in a novel Sobolev space relating to the diffusional driving forces, and prove convergence of a structure-preserving finite element discretization. This represents some of the first rigorous numerics for the coupling of multicomponent molecular diffusion with compressible convective flow. The broad applicability of our theory is illustrated with simulations of the centrifugal separation of noble gases and the microfluidic mixing of hydrocarbons.