Post-Doctoral Research Visit F - M Design Of Well-Balanced And Asymptotic-Preserving Methods In a High-Order Accurate Finite Difference Framework H/F INRIA

Post-Doctoral Research Visit F/M Design of well-balanced and asymptotic-preserving methods in a high-order accurate finite difference framework
Le descriptif de l'offre ci-dessous est en Anglais
Type de contrat : CDD

Niveau de diplôme exigé : Thèse ou équivalent

Fonction : Post-Doctorant

Contexte et atouts du poste

In the last three years, the structural method has been introduced to build very high-order numerical schemes to solve partial differential equations (PDEs) on compact stencils [1, 2]. A particularity of this finite difference method is that it approximates the solution and its derivatives with the same order of accuracy. It relies on defining two independent sets of discrete equations, the physical and the structural equations. The physical equations (PEs) describe the physics of the problem, i.e. the underlying PDEs. As such, treating problems with specific constraints (for instance, ensuring that some vector field is divergence-free) becomes a matter of adding or modifying a physical equation. The structural equations (SEs) are responsible for the accuracy of the discretization, and thus their modification makes it possible to treat non-smooth solutions or improve the accuracy of continuous ones.

Therefore, the structural method is a very general framework that offers a high degree of flexibility to construct high-order accurate schemes on compact stencils. Indeed, this flexibility comes from the separation between
physical and structural equations.

The physical equations can be modified to include constraints, or to adapt to the problem under consideration. The structural equations can be modified to control the accuracy and robustness of the scheme. For instance, consider a problem with a discontinuous solution. The structural equations can then be modified to lower the order of accuracy of the scheme, thus eliminating non-physical oscillations or increasing the stability of the method. This can be done by deriving the structural equations such that the scheme is exact on a piecewise linear solution, for instance.

The ANR project SMEAGOL aims at extending the structural method to the approximation of hyperbolic systems of balance laws in at least two spatial dimensions. For such problems, continuous initial conditions can generate non-smooth solutions in finite time, and multiscale regime changes frequently occur. This would make the method suitable to be used in problems from, for instance, equations governing problems in fluid mechanics or electromagnetism. Although the structural method is a general finite difference framework, it is particularly well-suited to such systems. Indeed, the separation between physics and discretization provides a natural setting to construct a scheme that can switch on or off physical and/or structural equations locally and on the fly, depending on the situation. SMEAGOL thus contains two sub-goals: the construction and the adaptation of the structural method.

[1] S.Clain, G.J.Machado, M.T.Malheiro. Compact schemes in time with applications to partial differential equations. Comput. Math. Appl. 140 (2023).
[2] S.Clain, R.M.S.Pereira, P.A.Pereira, D.Lopes. Structural schemes for one dimension stationary equations. Appl. Math. Comput. 457 (2023).

Mission confiée

The construction of the structural method is the focus of this postdoc project. It will be dedicated to the development of the structural method for hyperbolic systems of balance laws, and more specifically to the design of the well-balanced and asymptotic-preserving properties in this method. Since the structural method has been introduced quite recently, these questions remain open, even though similar ones have already been addressed in the literature for traditional finite difference or finite volume methods.

Breaking down this postdoc project, two sub-tasks are considered, each dedicated to a specific property to be satisfied by the resulting scheme.

- To simulate most real-world phenomena, an important requirement for a numerical method is to provide a good (or exact) approximation of stationary solutions. Well-balanced (WB) methods are crucial to describe accurately near-equilibrium solutions. For instance, considering a tsunami about to impact a coast, the water is almost at rest far from the coast. The tsunami simulation will be unusable unless the method provides a good approximation of this stationary solution. Obtaining the WB property translates to exactly discretizing some PDE, which usually comes with associated constraints. Common examples are divergence-free or curl-free constraints, widely present in fluid mechanics. As mentioned before, such constraints can be naturally incorporated in the structural method by adding a new physical equation.
As a consequence, the first task of this postdoc project is to implement a WB version of the structural method, generic enough to be applied to any system of balance laws.
- A usual requirement for numerical methods applied to hyperbolic systems of balance laws is to correctly approximate multiscale flows. In such cases, non-well-suited schemes can induce spurious instabilities. Handling multiscale phenomena is at the core of asymptotic-preserving (AP) schemes. For most hyperbolic systems of balance laws, two main multiscale regimes of interest arise. Both are connected to the singular limit of system parameters and the equations potentially losing hyperbolicity.
The first regime concerns the transition between compressible and incompressible flow. In the latter, divergence-free constraints play an important role.
The second one is related to stiff source terms, which lead to a regime transition from hyperbolic to parabolic, where additional constraints have to be satisfied.
Consequently, the second task of this postdoc project is to derive an AP version of the structural method, starting with the second limit systems (representing a transition from hyperbolic to parabolic).

Principales activités

Main activities :

- conducting a literature review of the structural method (and WB/AP methods if the applicant is unfamiliar with them)
- designing a WB structural method
- designing an AP structural method

Additional activities :

- implementating the method in an open-source code
- writing scientific articles and presenting the work at conferences

Compétences

- A strong experience in numerical analysis is expected (e.g., finite difference, finite volume or finite element methods). Knowledge of and experience in WB and AP methods is a plus.
- A background in hyperbolic partial differential equations would be appreciated.
- Coding skills are also essential. The code would be written in Python unless arguments are made for another language.

Avantages

- Subsidized meals
- Partial reimbursement of public transport costs
- Leave: 7 weeks of annual leave + 10 extra days off due to RTT (statutory reduction in working hours) + possibility of exceptional leave (sick children, moving home, etc.)
- Possibility of teleworking (after 6 months of employment) and flexible organization of working hours
- Professional equipment available (videoconferencing, loan of computer equipment, etc.)
- Social, cultural and sports events and activities
- Access to vocational training
- Social security coverage

Rémunération

€ 2788 gross/month