Quantum Computations of Partial Differential Equations

• Shi Jin (SJTU, Shanghai)

Room: Amphi Lebesgue Quantum computers are designed based on quantum mechanics principle, they are most suitable to solve the Schrodinger equation, and linear PDEs (and ODEs) evolved by unitary operators. It is important to to explore whether other problems in scientific computing, such as ODEs, PDEs, and linear algebra that arise in both classical and quantum systems which are not unitary evolution, can be handled by quantum computers. We will present a systematic way to develop quantum simulation algorithms for general differential equations. Our basic framework is dimension lifting, that transfers non-autonomous ODEs/PDEs systems to autonomous ones, nonlinear PDEs to linear ones, and linear ones to Schrodinger type PDEs—coined “Schrodingerization”—with unitary evolutions. Our formulation allows both qubit and qumode (continuous-variable) formulations, and their hybridizations, and provides the foundation for analog quantum computing which are easier to realize in the near term. We will also present dimension lifting techniques for quantum simulation of stochastic DEs and PDEs with fractional derivatives, and quantum machine learning. A quantum simulation software—“UnitaryLab”—will also be introduced.

Stability of the plasma sheath in a bounded interval

• Mehdi Badsi (Université de Nantes)

Room: Amphi Lebesgue In this talk, I will expose a stability study of plasma sheath equilibria for a one dimensional bounded Vlasov-Poisson type system.

Discrete hypocoercivity for a nonlinear kinetic reaction model

• Marianne Bessemoulin-Chatard (CNRS)

Room: Amphi Lebesgue In this talk, I will present a finite volume discretization of a 1D nonlinear kinetic reaction model, which describes a two-species recombination-generation process. More specifically, we establish the long-time convergence of the approximate solutions to equilibrium, at an exponential rate. To do this, we adapt the proof proposed in [Favre, Pirner, Schmeiser, ARMA 2023], based on an adaptation of the hypocoercivity method of [Dolbeault, Mouhot, Schmeiser, Trans. Amer. Math. Soc. 2015]. As in the continuous setting, this result is valid for bounded initial data and requires establishing a maximum principle, which necessitates the use of monotonic numerical fluxes. This is a joint work with Tino Laidin (Univ. Brest) and Thomas Rey (Univ. Nice).

High-order finite volume schemes and entopy inequalities

• Christophe Berthon (Université de Nantes)

Room: Amphi Lebesgue This work concerns the numerical approximations of the weak solutions of scalar hyperbolic conservation laws. After showing how to bypass the discrete entropy inequality barrier theorems for the linear advection, the derivation of a second-order entropy-satisfying scheme is presented for non-linear equations. The fully discrete stability result is established for regular strictly convex entropy and under a parabolic CFL-like condition. Some numerical experiments are done to assess the accuracy and the stability of the proposed scheme.