Ansgar Jüngel

Jan-Frederik Mennemann, Ansgar Jüngel, Hans Kosina

Transient simulations of a resonant tunneling diode oscillator are presented. The semiconductor model for the diode consists of a set of time-dependent Schrödinger equations coupled to the Poisson equation for the electric potential. The one-dimensional Schrödinger equations are discretized by the finite-difference Crank-Nicolson scheme using memory-type transparent boundary conditions which model the injection of electrons from the reservoirs. This scheme is unconditionally stable and reflection-free at the boundary. An efficient recursive algorithm due to Arnold, Ehrhardt, and Sofronov is used to implement the transparent boundary conditions, enabling simulations which involve a very large number of time steps. Special care has been taken to provide a discretization of the boundary data which is completely compatible with the underlying finite-difference scheme. The transient regime between two stationary states and the self-oscillatory behavior of an oscillator circuit, containing a resonant tunneling diode, is simulated for the first time.

Mario Bukal, Etienne Emmrich, Ansgar Jüngel

Structure-preserving numerical schemes for a nonlinear parabolic fourth-order equation, modeling the electron transport in quantum semiconductors, with periodic boundary conditions are analyzed. First, a two-step backward differentiation formula (BDF) semi-discretization in time is investigated. The scheme preserves the nonnegativity of the solution, is entropy stable and dissipates a modified entropy functional. The existence of a weak semi-discrete solution and, in a particular case, its temporal second-order convergence to the continuous solution is proved. The proofs employ an algebraic relation which implies the G-stability of the two-step BDF. Second, an implicit Euler and q-step BDF discrete variational derivative method are considered. This scheme, which exploits the variational structure of the equation, dissipates the discrete Fisher information (or energy). Numerical experiments show that the discrete (relative) entropies and Fisher information decay even exponentially fast to zero.

Francesca Bonizzoni, Marcel Braukhoff, Ansgar Jüngel et al.

An implicit Euler discontinuous Galerkin scheme for the Fisher-Kolmogorov-Petrovsky-Piscounov (Fisher-KPP) equation for population densities with no-flux boundary conditions is suggested and analyzed. Using an exponential variable transformation, the numerical scheme automatically preserves the positivity of the discrete solution. A discrete entropy inequality is derived, and the exponential time decay of the discrete density to the stable steady state in the L1 norm is proved if the initial entropy is smaller than the measure of the domain. The discrete solution is proved to converge in the L2 norm to the unique strong solution to the time-discrete Fisher-KPP equation as the mesh size tends to zero. Numerical experiments in one space dimension illustrate the theoretical results.

Ansgar Jüngel, Oliver Leingang

A fully discrete Galerkin scheme for a thermodynamically consistent transient Max-well-Stefan system for the mass particle densities, coupled to the Poisson equation for the electric potential, is investigated. The system models the diffusive dynamics of an isothermal ionized fluid mixture with vanishing barycentric velocity. The equations are studied in a bounded domain, and different molar masses are allowed. The Galerkin scheme preserves the total mass, the nonnegativity of the particle densities, their boundedness, and satisfies the second law of thermodynamics in the sense that the discrete entropy production is nonnegative. The existence of solutions to the Galerkin scheme and the convergence of a subsequence to a solution to the continuous system is proved. Compared to previous works, the novelty consists in the treatment of the drift terms involving the electric field. Numerical experiments show the sensitive dependence of the particle densities and the equilibration rate on the molar masses.

Claire Chainais-Hillairet, Ansgar Jüngel, Polina Shpartko

An implicit Euler finite-volume scheme for a spinorial matrix drift-diffusion model for semiconductors is analyzed. The model consists of strongly coupled parabolic equations for the electron density matrix or, alternatively, of weakly coupled equations for the charge and spin-vector densities, coupled to the Poisson equation for the elec-tric potential. The equations are solved in a bounded domain with mixed Dirichlet-Neumann boundary conditions. The charge and spin-vector fluxes are approximated by a Scharfetter-Gummel discretization. The main features of the numerical scheme are the preservation of positivity and L $\infty$ bounds and the dissipation of the discrete free energy. The existence of a bounded discrete solution and the monotonicity of the discrete free energy are proved. For undoped semiconductor materials, the numerical scheme is uncon-ditionally stable. The fundamental ideas are reformulations using spin-up and spin-down densities and certain projections of the spin-vector density, free energy estimates, and a discrete Moser iteration. Furthermore, numerical simulations of a simple ferromagnetic-layer field-effect transistor in two space dimensions are presented.

Anita Gerstenmayer, Ansgar Jüngel

A structure-preserving implicit Euler finite-element scheme for a degenerate cross-diffusion system for ion transport is analyzed. The scheme preserves the nonnegativity and upper bounds of the ion concentrations, the total relative mass, and it dissipates the entropy (or free energy). The existence of discrete solutions to the scheme and their convergence towards a solution to the continuous system is proved. Numerical simulations of two-dimensional ion channels using the finite-element scheme with linear elements and an alternative finite-volume scheme are presented. The advantages and drawbacks of both schemes are discussed in detail.

Claire Chainais-Hillairet, Ansgar Jüngel, Stefan Schuchnigg

The time decay of fully discrete finite-volume approximations of porous-medium and fast-diffusion equations with Neumann or periodic boundary conditions is proved in the entropy sense. The algebraic or exponential decay rates are computed explicitly. In particular, the numerical scheme dissipates all zeroth-order entropies which are dissipated by the continuous equation. The proofs are based on novel continuous and discrete generalized Beckner inequalities. Furthermore, the exponential decay of some first-order entropies is proved in the continuous and discrete case using systematic integration by parts. Numerical experiments in one and two space dimensions illustrate the theoretical results and indicate that some restrictions on the parameters seem to be only technical.

Clément Cancès, Claire Chainais-Hillairet, Anita Gerstenmayer et al.

An implicit Euler finite-volume scheme for a degenerate cross-diffusion system describing the ion transport through biological membranes is analyzed. The strongly coupled equations for the ion concentrations include drift terms involving the electric potential, which is coupled to the concentrations through the Poisson equation. The cross-diffusion system possesses a formal gradient-flow structure revealing nonstandard degen-eracies, which lead to considerable mathematical difficulties. The finite-volume scheme is based on two-point flux approximations with "double" upwind mobilities. It preserves the structure of the continuous model like nonnegativity, upper bounds, and entropy dis-sipation. The degeneracy is overcome by proving a new discrete Aubin-Lions lemma of "degenerate" type. Under suitable assumptions, the existence and uniqueness of bounded discrete solutions, a discrete entropy inequality, and the convergence of the scheme is proved. Numerical simulations of a calcium-selective ion channel in two space dimensions indicate that the numerical scheme is of first order.

Marianne Bessemoulin-Chatard, Ansgar Jüngel

A finite volume scheme for the (Patlak-) Keller-Segel model in two space dimensions with an additional cross-diffusion term in the elliptic equation for the chemical signal is analyzed. The main feature of the model is that there exists a new entropy functional yielding gradient estimates for the cell density and chemical concentration. The main features of the numerical scheme are positivity preservation, mass conservation, entropy stability, and - under additional assumptions - entropy dissipation. The existence of a discrete solution and its numerical convergence to the continuous solution is proved. Furthermore, temporal decay rates for convergence of the discrete solution to the homogeneous steady state is shown using a new discrete logarithmic Sobolev inequality. Numerical examples point out that the solutions exhibit intermediate states and that there exist nonhomogeneous stationary solutions with a finite cell density peak at the domain boundary.

Marianne Bessemoulin-Chatard, Claire Chainais-Hillairet, Ansgar Jüngel

We establish uniform L $\infty$ bounds for approximate solutions of the drift-diffusion system for electrons and holes in semiconductor devices, computed with the Schar-fetter-Gummel finite-volume scheme. The proof is based on a Moser iteration technique adapted to the discrete case.

Ansgar Jüngel, Stefan Schuchnigg

The exponential decay of the relative entropy associated to a fully discrete porous-medium equation in one space dimension is shown by means of a discrete Bakry-Emery approach. The first ingredient of the proof is an abstract discrete Bakry-Emery method, which states conditions on a sequence under which the exponential decay of the discrete entropy follows. The second ingredient is a new nonlinear summation-by-parts formula which is inspired by systematic integration by parts developed by Matthes and the first author. Numerical simulations illustrate the exponential decay of the entropy for various time and space step sizes.

Ansgar Jüngel, Panchi Li, Zhiwei Sun et al.

We propose a Bernoulli phase-fitted (BPF) finite difference method for the Helmholtz equation on the interval $(0, L)$ with impedance boundary conditions. The scheme is derived from a complexified Scharfetter--Gummel discretization of the one-way factorization of the Helmholtz operator. It yields both a phase-fitted interior discretization and exact discrete impedance boundary closures. For the homogeneous problem, the method is exact for plane waves, so the scheme introduces neither numerical dispersion in the interior nor artificial reflection at the boundaries. For the inhomogeneous problem, we prove well-posedness, derive wavenumber-explicit stability estimates, and establish second-order consistency and convergence valid for all $kh\notinπ\mathbb Z$, where $k$ is the wavenumber and $h$ the grid size. In particular, under the fixed-resolution condition $kh\le s_0$ for some $0<s_0<π$ together with $kL\geπ$, the constants in the error bounds remain uniform with respect to the wavenumber, yielding a pollution-free convergence theory in the principal Nyquist regime. Numerical experiments confirm the theoretical analysis and show favorable performance compared with standard and dispersion-corrected finite difference methods.

Bertram Düring, Philipp Fuchs, Ansgar Jüngel

A nonlinear diffusion equation, interpreted as a Wasserstein gradient flow, is numerically solved in one space dimension using a higher-order minimizing movement scheme based on the BDF (backward differentiation formula) discretization. In each time step, the approximation is obtained as the solution of a constrained quadratic minimization problem on a finite-dimensional space consisting of piecewise quadratic basis functions. The numerical scheme conserves the mass and dissipates the $G$-norm of the two-step BDF time approximation. Numerically, also the discrete entropy and variance are decaying. The decay turns out to be exponential in all cases. The corresponding decay rates are computed numerically for various grid numbers.

Ansgar Jüngel, Stefan Schuchnigg

Semi-discrete Runge-Kutta schemes for nonlinear diffusion equations of parabolic type are analyzed. Conditions are determined under which the schemes dissipate the discrete entropy locally. The dissipation property is a consequence of the concavity of the difference of the entropies at two consecutive time steps. The concavity property is shown to be related to the Bakry-Emery approach and the geodesic convexity of the entropy. The abstract conditions are verified for quasilinear parabolic equations (including the porous-medium equation), a linear diffusion system, and the fourth-order quantum diffusion equation. Numerical experiments for various Runge-Kutta finite-difference discretizations of the one-dimensional porous-medium equation show that the entropy-dissipation property is in fact global.