Section: New Results

Hybrid time discretizations of high-order

High order time discretization for dissipative wave equations.

Participants : Juliette Chabassier, Julien Diaz, Anh-Tuan Ha, Sébastien Imperiale.

Magique-3D team is interested in numerical methods for wave propagation in realistic media, which are naturally dissipative in many application cases. In this internship, we wish to investigate several dissipation models, that lead to Partial Differential Equations with different structures. The simplest model is the scalar wave equation with homogeneous and constant damping 2ut2+Rut-Δu=f. In order to approach the complexity of the propagating medium and its geometry, high order finite elements in space are used. Once the spatial discretization is fixed, we get a differential equation of the kind d2uhdt2+Bhduhdt+Ahuh=fh, where the mass matrix is the identity thanks to the mass lumping technique followed by a renormalization, Bh is the dissipation matrix and Ah the stiffness matrix. Classically, this equation is discretized in time with a centered and second order finite difference scheme known as the θ-scheme (θ>0)

u h n + 1 - 2 u h n - u h n - 1 Δ t 2 + B h u h n + 1 - u h n - 1 2 Δ t + A h θ u h n + 1 + ( 1 - 2 θ ) u h n + θ u h n - 1 = f h n (1)

In order to preserve the precision obtained with high order finite elements in space, we wish to design higher order time discretizations, while preserving some interesting mathematical properties as the dissipation of a discrete energy, and an efficiency close to the one observed for the second order scheme. More precisely, if θ=0 and Bh is diagonal, scheme (1) only requires the inversion of a diagonal matrix at each time step.

We want to use the technique of the modified equation, which consists in compensating the first term of the consistency error of a low order discretization, by adding a well chosen new term. If θ=0, this approach leads to the following fourth order accurate in time scheme

I h + Δ t 2 12 B h u h n + 1 - 2 u h n - u h n - 1 Δ t 2 + B h + Δ t 2 12 B h A h - A h B h u h n + 1 - u h n - 1 2 Δ t + A h u h n - Δ t 2 12 A h 2 u h n = f ˜ h n

Even if Bh is diagonal, Ah and Bh do not commute in general. We propose to replace the matrix BhAh-AhBh, potentially hard to invert, by an approximated matrix, easy to invert, without deteriorating the consistency of the scheme.

An article is being written and will be submitted soon.

High order conservative explicit and implicit schemes for wave equations.

Participants : Juliette Chabassier, Sébastien Imperiale.

In 2016 we have studied the space/time convergence of a family of high order conservative explicit and implicit schemes for wave equations. An original proof of convergence has been proposed and provides an understanding of the lack of convergence of some schemes when the time step approaches its greatest admissible value for stability (CFL condition). An article has been submitted.

Efficient high order implicit time schemes for Maxwell's equations.

Participants : Hélène Barucq, Marc Duruflé, Mamadou N'Diaye.

The Padé approximant is well known to be one of the best approximation of an exponential function which is involved in the exact solution of the linear ODE (Ordinary Differential Equations):

y ' ( t ) = A y ( t ) + F ( t )

where A is a given matrix (usually coming from finite element discretization) and F is a term source. The numerical solution can be constructed by approximating the exponential function using the diagonal Padé approximant:

R ( z ) = P m ( z ) Q m ( z )

The function R(z) is a fraction involving two polynomial Pm and Qm of same degree and approximating the exponential. The corresponding scheme is implicit and A-stable in the sense of Dahlquist. The associated stability function is the same as the stability function of the Gauss-Runge-Kutta schemes. However, Gauss-Runge-Kutta schemes can be used to handle non-linear ODEs, but they are too expensive to use in practice. The diagonal Padé schemes presented here can be seen as a simplification of Gauss-Runge-Kutta schemes in the case of linear ODE. We have proposed an efficient way to implement the diagonal Padé schemes with an accurate approximation of the source term to keep the correct order of accuracy.

The main drawback of Padé schemes is that the denominator Qm(z) has distinct roots. It implies that we have to solve distinct linear systems at each time step. As a result, we have also studied the case where the denominator has an unique real root γ :

R ( z ) = N ( z ) ( 1 - γ z ) m

The numerator N is then found to obtain the "best" approximation of the exponential under the constraint of the A-stability property of the underlying schemes. The obtained schemes have been called Linear Singly Diagonal Implicit Runge-Kutta schemes (Linear SDIRK) since they share the same property as SDIRK (a unique linear system to solve several times) but they can be applied only to linear ODEs. We provide a performance assessment of different implicit schemes (Padé schemes, SDIRK and Linear SDIRK). The comparison criteria are based on the amplitude and phase errors which are reliable gauges of accuracy when approximating waves problems. The Linear SDIRK schemes and the diagonal Padé schemes have been implemented in the code Montjoie. We have performed numerical experiments in 1-D and 2-D for Maxwell's equations to validate these schemes and compare their efficiency.

This work has been presented at the conference ICOSAHOM  [28], the colloquium Inter'Actions en Mathématiques Lyon 2016 and the Mathias annual Total seminar  [27].

Optimized high-order explicit Runge-Kutta-Nyström schemes.

Participants : Marc Duruflé, Mamadou N'Diaye.

In this work we propose a high order time integration explicit scheme to solve a second order derivative non-linear ordinary differential equation (ODE)

y ' ' = f ( t , y )

To solve this family of ODEs, explicit one-step Runge-Kutta-Nyström have been proposed by Hairer et al. The stability condition (CFL) associated with these schemes have been studied for order 3, 4 and 5 by Chawla and Sharma. In this work, we have extended the stability studies for high order. We proposed optimal coefficients for Runge-Kutta-Nyström schemes of order 6, 7, 8 and 10 which have been obtained by optimizing the CFL. With the obtained optimal CFL, these schemes are well suited for stiff problems where the stability condition is restrictive. These schemes have been implemented in the code Montjoie.

Numerical experiments have been conducted in 1-D for the non-linear Maxwell's equation and show that obtained Runge-Kutta-Nyström schemes of order 7 is quite efficient. This work has been presented at the conference ICOSAHOM [48].