# Innovative numerical methods for evolutionary partial differential equations and applications

Università degli Studi di ROMA "La Sapienza", Consiglio Nazionale delle Ricerche, Università degli Studi di FERRARA, Università degli Studi di TRENTO, Università degli Studi di VERONA

## Abstract

The main goal of the project consists in the development and analysis of new numerical methods for problems governed by hyperbolic systems of partial differential equations (PDE) with applications to various fields. Specifically, the main research topics of the project are the following:

A1 Asymptotic preserving (AP) methods for multiscale problems

A2 High-order semi-Lagrangian (SL) schemes

A3 Well balanced (WB) and structure preserving (SP) schemes

A4 Optimal control problems and mean-field games (MFG)

A5 Dispersive and highly oscillatory waves

A6 Uncertainty Quantification (UQ)

A7 High performance computational fluid dynamics

A8 Modeling and simulation of traffic flow and crowd dynamics

Many physical systems involve multiple space and time scales. Typical examples are low Mach number flows, rarefied gas dynamics with small (but not negligible) Knudsen number, the Schrödinger equation near the semi-classical limit. In all these cases there is a small parameter that identifies the ratio between typical spatial or temporal scales. When the parameter vanishes, the system may be described by a reduced set of equations. For example, as Mach number vanishes, gas flow is described by the incompressible Navier-Stokes (NS) equations. For small Knudsen numbers the Boltzmann equation reduces to the compressible Euler or NS equations, and when the scaled Planck constant vanishes, quantum mechanics reduces to classical mechanics.

Direct numerical solutions that resolve all scales become prohibitively expensive in most cases, hence it is mandatory to capture the limit behavior with under-resolved schemes that still provide accurate solutions, but with much greater efficiency. Numerical methods with this property are called Asymptotic Preserving (AP) [Jin99, Jin12].

The development of efficient AP schemes for multi scale problems is the main underlying theme of the project (A1). This goal can be achieved with a variety of techniques, e.g. methods that allow large time steps (semi-Lagrangian methods (A2), implicit-explicit [IMEX] ODE solvers (A1), exponential methods (A1,A5), and so on) or adapt to complex multiscale geometry (adaptive mesh refinement). In some cases, a reformulation of the problem is needed, in order to capture the limit behavior without resolving the small scales (A5).

Particular attention is devoted to the construction of numerical methods that maintain some properties of the PDE system at the discrete level (A3). This is relevant for the Boltzmann equation in kinetic models, for which collision invariants guarantee conservation of mass, momentum and energy at the microscopic level, or for magneto hydrodynamics (MHD), where the solenoidal property of the magnetic field is maintained by the equations.

The construction of efficient schemes for optimal control problems of large systems of differential equations will also be considered (A4). A related subject involves mean-field games composed by a Hamilton–Jacobi–Bellman equation coupled with a Fokker–Planck equation.

In most applications, the parameters defining the model are not precisely known. It is therefore crucial to investigate the effect of the uncertainty in the data on the solution (A6).

The last themes (A7,A8) are close to real world problems.

In this ambitious project we aim at making innovative contributions in applied mathematics and scientific computing with new schemes that go beyond the state of the art and that preserve as much as possible of the structure of the underlying continuous equations also at the discrete level. The applicability will be demonstrated on a set of prototype applications, also making use of modern high performance computing (HPC).

The proposal is structured as follows.
In B.1-2 we state the main objectives subdivided into 9 work packages, put them in perspective with the literature and describe the methodologies we plan to adopt for each of them. Sec. B.1-3 describes a time planning, emphasizing the complementary competences and collaborations of the various unit members. Sec. B.1-4 illustrates the outcome of the project in terms of applications to various fields and high level education.

Particular care will be given to the formation of young researchers (PhD students and post-docs), by deeply involving them in the research project, and exploiting the synergy with other European projects of the PIs in related areas, such as the ETN-ITN network ModCompShock (http://modcompshock.eu), the H2020 FET-HPC project ExaHyPE (http://exahype.eu/) or the French-German-Italian Laboratoire International Associé (LIA).

In part B references are indicated as follows:
references included in the 116 publications of the unit coordinators are given as Xx.n, with n in [1..20], and Xx the first two letters of the coordinator surname (e.g. Ru = Russo, Pa = Pareschi).
Additional references are reported in section B.1.6.