Towards a new approach of mesh adaptation methods and its impact on accuracy for the simulation of stiff PDEs

Marc Massot

The present work is the result of a team work involving

• Thomas Bellotti (CNRS, EM2C - Fédé Maths CS)
• Loïc Gouarin (École polytechnique, CMAP)
• Josselin Massot (École polytechnique, CMAP)
• Pierre Matalon (École polytechnique, CMAP)
• Laurent Séries (École polytechnique, CMAP)
• Christian Tenaud (CNRS, EM2C - Fédé Maths CS)

website of the team

https://github.com/hpc-maths/

The present work is the result of a team work involving

• Thomas Bellotti (CNRS, EM2C - Fédé Maths CS)
• Loïc Gouarin (IR École polytechnique, CMAP)
• Josselin Massot (IR École polytechnique, CMAP)
• Pierre Matalon (IR École polytechnique, CMAP)
• Laurent Séries (IR École polytechnique, CMAP)
• Christian Tenaud (CNRS, EM2C - Fédé Maths CS)

website of the team

https://github.com/hpc-maths/

Context

Burgers equation - small hat problem

\[ \partial_t u + \partial_x \left ( f(u) \right ) = 0, \quad t \geq 0, \quad x \in \mathbb{R}, \qquad f(u) = \dfrac{u^2}{2}, \]

Consider the Cauchy problem with initial cond.

\[ u^0(x)=\left\{% \begin{array}{cc} \hfill 0, & x \in ]- \infty,-1]\cup [1, + \infty[,\\ x+1, & x \in ]-1,0], \hfill \\ 1-x, & x \in[0,1[. \hfill \\ \end{array} \right. \]

• Shock formation at time \(T^* = 1\)
• Leading to irreversible solution
• RH condition governs shock dynamics

Burgers equation - small hat problem

\[ \partial_t u + \partial_x \left ( f(u) \right ) = 0, \quad t \geq 0, \quad x \in \mathbb{R}, \qquad f(u) = \dfrac{u^2}{2}, \]

Consider the Cauchy problem with initial cond.

\[ u^0(x)=\left\{% \begin{array}{cc} \hfill 0, & x \in ]- \infty,-1]\cup [1, + \infty[,\\ x+1, & x \in ]-1,0], \hfill \\ 1-x, & x \in[0,1[. \hfill \\ \end{array} \right. \]

• Shock formation at time \(T^* = 1\)
• Leading to irreversible solution
• RH condition governs shock dynamics

Burgers equation - small hat problem

\[ \partial_t u + \partial_x \left ( f(u) \right ) = 0, \quad t \geq 0, \quad x \in \mathbb{R}, \qquad f(u) = \dfrac{u^2}{2}, \]

Consider the Cauchy problem with initial cond.

\[ u^0(x)=\left\{% \begin{array}{cc} \hfill 0, & x \in ]- \infty,-1]\cup [1, + \infty[,\\ x+1, & x \in ]-1,0], \hfill \\ 1-x, & x \in[0,1[. \hfill \\ \end{array} \right. \]

• Shock location \(\varphi(t)=\sqrt{2(1+t)}-1\)
• Propagation speed shock \(\sigma(t)={1}/{\sqrt{2(1+t)}}\)
• Shock amplitude \([u]=\sqrt{{2}/{(1+t)}}\)

The key issue is to get a analytical solution of the problem involving both reversible and irreversible dynamics. Notebook MAP412 pour intégration Godunov et convergence https://jupyter_map412.gitlab.labos.polytechnique.fr/jb_2024_2025/content/chapter/11_edp_03/burgers.html

Burgers equation - sinus problem

\[ \partial_t u + \partial_x \left ( f(u) \right ) = 0, \quad t \geq 0, \quad x \in \mathbb{R}, \qquad f(u) = \dfrac{u^2}{2}, \]

Consider the Cauchy problem with initial conditions:

\[ u^0(x) = \frac{1}{2} (1+\sin(\pi(x-1))) \quad x \in [-1,1] \]

Adaptive Multiresolution

• Minimum level \(\underline{\ell}\) and maximum level \(\bar{\ell}\).
• Cells: \[ C_{\ell, k}:=\prod_{\alpha=1}^d\left[2^{-\ell} k_\alpha, 2^{-\ell}\left(k_\alpha+1\right)\right] \]
• Finest step: \(\Delta x=2^{-\bar{\ell}}\).
• Level-wise step: \(\Delta x_{\ell}:=2^{-\ell}=2^{\Delta \ell} \Delta x\).

Just introducing the notations and the notion of level difference with Delta l

Wavelets

Decomposition of the solution on a wavelet basis [Daubechies, ’88], [Mallat, ’89] to measure its local regularity. “Practical” approach by [Harten, ’95], [Cohen et al., ’03], [Duarte et al., ’11-’16].

Projection operator

Prediction operator at order \(2 s +1\)

\[ {\hat f}_{\ell+1,2 k}={f}_{\ell, k}+\sum_{\sigma=1}^s \psi_\sigma\left({f}_{\ell, k+\sigma}-{f}_{\ell, k-\sigma}\right) \]

Details are regularity indicator \[ {\mathrm{d}}_{\ell, {k}}:={f}_{\ell, {k}}-{\hat{f}}_{\ell, {k}} \]

Let \(f \in W^{\nu, \infty}\) (neigh. of \(C_{\ell, k}\) ), then \[ \left|{\mathrm{d}}_{\ell, k}\right| \lesssim 2^{-\ell \min (\nu, 2 s +1)}|f|_{W^{\min (\nu, 2 s +1), \infty}} \]

Fast wavelet transform:

means at the finest level can be recast as means at the coarsest level + details \[ \begin{array}{rlr} {f}_{\overline{\ell}} & \Longleftrightarrow & \left({f}_{\underline{\ell}}, {{d}}_{\underline{\ell} +1}, \ldots, {d}_{\bar{\ell}}\right)\\ \end{array} \]

Mesh coarsening (static)

Local regularity of the solution allows to select areas to coarsen

\[ {{f}}_{\bar{\ell}} \rightarrow \left({f}_{\underline{\ell}}, {\mathbf{d}}_{\underline{\ell}+1}, \ldots, {\mathbf{d}}_{\bar{\ell}}\right) \rightarrow \left({f}_{\underline{\ell}}, {\tilde{\mathbf{d}}}_{\underline{\ell}+1}, \ldots, \tilde{{\mathbf{d}}}_{\bar{\ell}}\right) \rightarrow {\tilde{{f}}}_{\bar{\ell}} \] \[ \tilde{{\mathrm{d}}}_{\ell, k}= \begin{cases}0, & \text { if } \left|{\mathbf{d}}_{\ell, k}\right| \leq \epsilon_{\ell}=2^{-d \Delta \ell} \epsilon, \quad \rightarrow \quad\left\|{\mathbf{f}}_{\bar{\ell}}-\tilde{{\mathbf{f}}}_{\bar{\ell}}\right\|_{\ell^p} \lesssim \epsilon \\ {\mathrm{d}}_{\ell, k}, & \text { otherwise} \end{cases} \]

Set a small (below \(\epsilon_{\ell}\)) detail to zero \(\equiv\) erase the cell \(C_{\ell, k}\) from the structure

Examples

Equation

\[ f(x) = exp(-50x^2) \; \text{for} \; x\in[-1, 1] \]

min level	1
max level	12
ε	10-3
compression rate	96.29%
error	0.00078

Examples

Equation

\[ f(x) = \left\{ \begin{array}{l} 1 - |2x| \; \text{if} \; -0.5 < x < 0.5,\\ 0 \; \text{elsewhere} \end{array} \right. \]

min level	1
max level	12
ε	10-3
compression rate	98.49%
error	0

Examples

Equation

\[ f(x) = 1 - \sqrt{\left| sin \left( \frac{\pi}{2} x \right) \right|} \; \text{for} \; x\in[-1, 1] \]

min level	1
max level	12
ε	10-3
compression rate	96.29%
error	0.00053

Examples

Equation

\[ f(x) = \tanh(50 |x|) - 1 \; \text{for} \; x\in[-1, 1] \]

min level	1
max level	12
ε	10-3
compression rate	97.46%
error	0.002

Time evolution of PDEs

• Finite volumes with global time step \(\Delta t = \Lambda(\Delta x)\)
• Use dynamic mesh refinement

Mesh updated using “old” information at time \(t\) to accommodate the one at time \(t + \Delta t\)

• Propagation of information : add security cells
• Formation of singularities : (regularity index: \(\nu =0\), \(\mu = \min(\nu,2 s +1)\)) refine if \[ \left|{\mathbf{d}}_{\ell, k}\right| \geq \epsilon_{\ell}\,2^{d+\mu} \]

Finite volumes / conservation / order

Flux evaluation at interfaces between levels

Using the prediction operator allows to evaluate fluxes at the same level

Finite volume method

We use a Godunov flux for the small hat problem

Numerical Analysis and Modified Equations

Linear scalar transport equation

In this work, we are concerned with the numerical solution of the Cauchy problem associated with the linear scalar conservation law

\[ \partial_t u(t, x)+V \partial_x u(t, x)=0, \quad(t, x) \in \mathbb{R}^{+} \times \mathbb{R} \]

where \(V\) is the transport velocity, taken \(V>0\) without loss of generality. we consider 1 d problems. The extension to \(2\mathrm{~d}\) / \(3\mathrm{~d}\) problems is straightforward and usually done by tensorization [Bellotti2022] and yields analogous conclusions.

The discrete volumes are \[ C_{\ell, k}:=\left[2^{-\ell} k, 2^{-\ell}(k+1)\right], \quad k \in \{ 0,2^{\ell}-1 \}, \] for any \(\ell \in \{ \underline{\ell}, \bar{\ell} \}\). The measure of each cell at level \(\ell\) is \(\Delta x_{\ell}:=2^{-\ell}\) and we shall indicate \(\Delta x:=\Delta x_{\bar{\ell}}\). The cell centers are \(x_{\ell, k}:=\) \(2^{-\ell}(k+1 / 2)\). Finally, we shall indicate \(\Delta \ell:=\bar{\ell}-\ell\), hence \(\Delta x_{\ell}=2^{\Delta \ell} \Delta x\).

Finite Volume scheme

Finite Volume scheme at the finest level of resolution \(\bar{\ell}\) for any cell of indices \(\bar{k} \in \{ 0,2^{\bar{\ell}}-1 \}\). Explicit schemes read:

\[ \mathrm{v}_{\bar{\ell}, \bar{k}}^{n+1}=\mathrm{v}_{\bar{\ell}, \bar{k}}^n-\frac{\Delta t}{\Delta x}\left(\Phi\left(\mathrm{v}_{\bar{\ell}, \bar{k}+1 / 2}^n\right)-\Phi\left(\mathrm{v}_{\bar{\ell}, \bar{k}-1 / 2}^n\right)\right) \]

where we utilize the same linear numerical flux for the left and the right flux (conservativity) \[ \Phi\left(\mathbf{v}_{\bar{\ell}, \bar{k}-1 / 2}\right):=V \sum_{\alpha=\underline{\alpha}}^{\bar{\alpha}} \phi_\alpha \mathbf{v}_{\bar{\ell}, \bar{k}+\alpha}, \quad \Phi\left(\mathbf{v}_{\bar{\ell}, \bar{k}+1 / 2}\right):=V \sum_{\alpha=\underline{\alpha}}^{\bar{\alpha}} \phi_\alpha v_{\bar{\ell}, \bar{k}+1+\alpha} \]

Modified equations

[Carpentier et al 97] or Cauchy-Kowalewski procedure [Harten et al 87]

\[ \partial_t u\left(t^n, x_{\bar{\ell}, \bar{k}}\right)+V \partial_x u\left(t^n, x_{\bar{\ell}, \bar{k}}\right)=\sum_{h=2}^{+\infty} \Delta x^{h-1} \sigma_h \partial_x^h u\left(t^n, x_{\bar{\ell}, \bar{k}}\right) \]

• Upwind scheme \[ \partial_t u+V \partial_x u=\frac{\Delta x V}{2}(1-\lambda V) \partial_{x x} u+O\left(\Delta x^2\right) \]
• Lax-Wendroff scheme \[ \partial_t u+V \partial_x u=-\frac{\Delta x^2 V}{6}\left(1-\lambda^2 V^2\right) \partial_x^3 u+O\left(\Delta x^3\right) \]
• OSMP-3 scheme \[ \partial_t u+V \partial_x u=\frac{\Delta x^3 V}{24}\left(-\lambda^3 V^2+2 \lambda^2 V^2+\lambda V-2\right) \partial_x^4 u+O\left(\Delta x^4\right) \]

How to include MRA I

We introduce the reconstruction operator \(\hat{s}\) instead of \(s\) on the cells \(\left(\bar{\ell}, 2^{\Delta \ell} k+\delta\right)\) for any \(\delta \in \mathbb{Z}\) at the finest level

• \(\hat{s}=s\) : exact local flux reconstruction [Cohen et al. 2003].
• \(\hat{s}=0\) but \(s>0\), direct evaluation or naive evaluation [Hovhannisyan et al 2010].

\[ \mathbf{w}_{\bar{\ell}, \bar{k}}^{n+1}=\mathbf{w}_{\bar{\ell}, \bar{k}}^n-\frac{\Delta t}{\Delta x}\left(\Phi\left(\hat{\hat{\mathbf{w}}}_{\bar{\ell}, \bar{k}+1 / 2}^n\right)-\Phi\left(\hat{\hat{\mathbf{w}}}_{\bar{\ell}, \bar{k}-1 / 2}^n\right)\right) \] \[ \Phi\left(\hat{\hat{\mathbf{w}}}_{\bar{\ell}, \bar{k}-1 / 2}\right):=V \sum_{\alpha=\underline{\alpha}}^{\bar{\alpha}} \phi_\alpha \hat{\hat{\mathbf{w}}}_{\bar{\ell}, \bar{k}+\alpha} \]

How to include MRA II

Let now \((\ell, k) \in S\left(\tilde{\Lambda}^{n+1}\right)\), taking the projection yields the multiresolution scheme

\[ \mathbf{w}_{\ell, k}^{n+1}=\mathbf{w}_{\ell, k}^n-\frac{\Delta t}{\Delta x_{\ell}}\left(\Phi\left(\hat{\hat{\mathbf{w}}}_{\bar{\ell}, 2^{\Delta \ell}(k+1)+1 / 2}^n\right)-\Phi\left(\hat{\hat{\mathbf{w}}}_{\bar{\ell}, 2^{\Delta \ell} k-1 / 2}^n\right)\right) \]

\[ \Phi\left(\hat{\hat{\mathbf{w}}}_{\bar{\ell}, 2^{\Delta \ell} k-1 / 2}\right):=V \sum_{\alpha=\underline{\alpha}}^{\bar{\alpha}} \phi_\alpha \hat{\hat{\mathbf{w}}}_{\bar{\ell}, 2^{\Delta \ell} k+\alpha} \]

Some information is loss because of the averaging procedure: two different schemes we can consider for the computation of the modified equations.

Theorem

The local truncation error of the reference Finite Volume scheme and the one of the adaptive Finite Volume scheme are the same up to order \(2\hat{s}+1\) included.

Modified equations including MRA

This result establishes at which order the modified equations of the reference scheme are perturbed by the introduction of the adaptive scheme. However, it does not characterize the terms in the modified equations above order \(2\hat{s}+1\) in \(\Delta x\) (symbolic computations).

• Upwind scheme (original) \[ {\color{blue}{% \partial_t u+V \partial_x u=\frac{\Delta x V}{2}(1-\lambda V) \partial_{x x} u+O\left(\Delta x^2\right) }} \]
• Upwind scheme with MRA \[ \begin{array}{lr} \partial_t u+V \partial_x u=\frac{\Delta x V}{2}\left(2^{\Delta \ell}-\lambda V\right) \partial_{x x} u+O\left(\Delta x^2\right), & \text { for } \hat{s}=0 \\ \partial_t u+V \partial_x u=\frac{\Delta x V}{2}(1-\lambda V) \partial_{x x} u-\frac{\Delta x^2 V}{6}\left(1-\lambda^2 V^2\right) \partial_x^3 u+& \\ \ \ \ \ \ \ \ \ \ \ \frac{\Delta x^3 V}{24}\left(-3 \Delta \ell 2^{2 \Delta \ell}+2^{2 \Delta \ell}-\lambda^3 V^3\right) \partial_x^4 u+O\left(\Delta x^4\right), & \text { for } \hat{s}=1 \end{array} \]

Modified equations including MRA

This result establishes at which order the modified equations of the reference scheme are perturbed by the introduction of the adaptive scheme. However, it does not characterize the terms in the modified equations above order \(2\hat{s}+1\) in \(\Delta x\) (symbolic computations).

• Lax-Wendroff scheme (original) \[ {\color{blue}{\partial_t u+V \partial_x u=-\frac{\Delta x^2 V}{6}\left(1-\lambda^2 V^2\right) \partial_x^3 u+O\left(\Delta x^3\right)}} \]
• Lax-Wendroff scheme with MRA \[ \begin{array}{lr} \partial_t u+V \partial_x u=\frac{\Delta x \lambda V^2}{2}\left(2^{\Delta \ell}-1\right) \partial_{x x} u+O\left(\Delta x^2\right), & \text { for } \hat{s}=0 \\ \partial_t u+V \partial_x u=-\frac{\Delta x^2 V}{6}\left(1-\lambda^2 V^2\right) \partial_x^3 u+&\\ \ \ \ \ \ \ \ \ \ \ \frac{\Delta x^3 \lambda V^2}{24}\left(-3 \Delta \ell 2^{2 \Delta \ell}+2^{2 \Delta \ell}-\lambda^2 V^2\right) \partial_x^4 u+O\left(\Delta x^4\right), & \text { for } \hat{s}=1 \end{array} \]

Theoretical results on the global error

Theorem 2

Assume that

• The reference scheme satisfies the restricted stability condition \(\|E\| \leq 1\)
• The Harten-like scheme satisfies the restricted stability condition \(\left\|\bar{E}_{\Lambda}\right\| \leq 1\) for any \(\Lambda\).

Then, for smooth solution, in the limit \(\Delta x \rightarrow 0\) (i.e. \(\bar{\ell} \rightarrow+\infty\) ) and for \(\Delta \underline{\ell}=\bar{\ell}-\underline{\ell}\) kept fixed, we have the error estimate

\[ \left\|\mathbf{v}_{\bar{\ell}}^n-\mathbf{w}_{\bar{\ell}}^n\right\| \leq C_{t r} t^n \Delta x^{2 \hat{s}+1}+C_{m r} \frac{t^n}{\lambda \Delta x} \epsilon \]

where \(C_{t r}=C_{t r}\left(\bar{\ell}-\underline{\ell},\left(\phi_\alpha\right)_\alpha, \lambda, \hat{s}, V\right)\) and \(C_{m r}=C_{m r}\left(\bar{\ell}-\underline{\ell},\left(\phi_\alpha\right)_\alpha, \lambda, \hat{s}, s, V\right)\). \[ \left\|\mathbf{u}_{\bar{\ell}}^n-\mathbf{w}_{\bar{\ell}}^n\right\| \leq C_{r e f} t^n \Delta x^\theta+C_{t r} t^n \Delta x^{2 \hat{s}+1}+C_{m r} \frac{t^n}{\lambda \Delta x} \epsilon \]

Let us start by discussing the assumption that we have placed in the statement of Theorem 2: - The restricted stability condition \(\|E\| \leq 1\) could be replaced by a milder condition \(\|E\| \leq 1+C \Delta t\) for some constant \(C \geq 0\), see (69) in [7] and (A2) in [15]. This would not change the result. The technical assumption \(\left\|\bar{E}_{\Lambda}\right\| \leq 1\) is harder to relax and also difficult to check in practice. - The fact of considering smooth solutions comes from the fact that we want to apply the analysis of the modified equations to obtain the convergence rates, in the spirit of the Lax theorem [1]. For the same reason, we take \(\Delta x \rightarrow 0\) (or \(\bar{\ell} \rightarrow+\infty\) ). - The distance between maximum and minimum level \(\Delta \underline{\ell}=\bar{\ell}-\underline{\ell}\) has to be fixed, because otherwise the constant \(C_{\text {tr }}\) potentially explodes and dominates \(\Delta x^{2 \hat{s}+1}\) when \(\Delta x \rightarrow 0\). This would prevent us from comparing orders. Moreover, this is also reasonable from the standpoint of actual computations, where we refine the mesh to achieve convergence (or nearly so) keeping the number of different available grid levels fixed. Still, we shall also perform numerical demonstration without fixing \(\Delta \underline{\ell}=\bar{\ell}-\underline{\ell}\) to show that the modified equations that we have previously developed provide important information on the behavior of \(\left\|\mathbf{v}_{\bar{\ell}}^n-\mathbf{w}_{\bar{\ell}}^n\right\|\).

Comments on theorem

• The error estimate contains three contributions: the discretization error of the reference scheme, the perturbation error between the reference and the adaptive scheme, and the thresholding error coming from the multiresolution
• The constant \(C_{\mathrm{tr}}\) generally grows exponentially with \(\bar{\ell}-\underline{\ell}\), sometimes also involving linear terms, i.e. \(\hat{s}=1\). We have the following cases:
  • \(\theta<2 \hat{s}+1\). The error of the reference scheme dominates the perturbation introduced by the adaptive scheme \(\left\|\mathbf{u}_{\bar{\ell}}^N-\mathbf{w}_{\bar{\ell}}^N\right\| \leq C_{\mathrm{ref}} T \Delta x^\theta+C_{\mathrm{mr}} \frac{T}{\lambda \Delta x} \epsilon\). A thresholding error of the same order as the reference error \(\epsilon \sim \Delta x^{\theta+1}\).
  • \(\theta=2 \hat{s}+1\). The error of the reference scheme and the perturbation order are comparable (first example!) We have \(\left\|\mathbf{u}_{\bar{\ell}}^N-\mathbf{w}_{\bar{\ell}}^N\right\| \leq\left(C_{\mathrm{ref}}+C_{\mathrm{tr}}\right) T \Delta x^\theta+C_{\mathrm{mr}} \frac{T}{\lambda \Delta x} \epsilon\).
  • \(\theta>2 \hat{s}+1\). The perturbation introduced by the adaptive scheme dominates the error of the reference scheme. Therefore, multiresolution introduces a large perturbation that yields a different convergence rate. We have \(\left\|\mathbf{u}_{\bar{\ell}}^N-\mathbf{w}_{\bar{\ell}}^N\right\| \leq C_{\mathrm{tr}} T \Delta x^{2 \hat{s}+1}+C_{\mathrm{mr}} \frac{T}{\lambda \Delta x} \epsilon\), thus \(\epsilon \sim \Delta x^{2 \hat{s}+2}\) (AMR !)

Assume that for the choice of \(\bar{\ell}-\underline{\ell}\) at hand, we have \(C_{\mathrm{tr}} \sim C_{\mathrm{ref}}\), then w

How to compute fluxes at the finest level

As we have seen, calculating the flux at the finest level is crucial to obtaining good error estimates. But how does this work in practice?

Let’s take a simple example.

How to compute fluxes at the finest level

Let’s imagine that our finest level is 4 levels away from where we are.

How to compute fluxes at the finest level

Now let’s imagine that we want to calculate the flow using this small cell on the left. We use an operator with s = 1, so we need 3 cells on the level below to calculate it. These 3 cells need 4 cells on the level below to be calculated. We can repeat the process indefinitely and we’ll still need 4 cells on the bottom level.

We can determine the coefficients of our linear application to go from our level to any cell on a higher level. There’s no need for an intermediate level.

How to compute fluxes at the finest level

We can do the same for the right-hand cell. We can see that the cells used are not exactly the same.

Now, if we wanted to reconstruct all the cells of the finest level, we’d only need these 5 cells, with different coefficients depending on the cell we’re trying to reconstruct.

Burgers results

Burgers results (Error for scheme order 1)

Burgers results (Error for scheme order 1)

Burgers results (Error for scheme order 1)

Burgers results (MR solution and level for scheme order 1)

Burgers results (MR solution and error for scheme order 1)

Burgers results (MR+MLF sol. and level for scheme order 1)

Burgers results (MR+MLF sol. and error for scheme order 1)

Burgers results (Error for scheme order 1)

Burgers results (MR solution and level for scheme order 1)

Burgers results (MR solution and error for scheme order 1)

Burgers results (MR+MLF sol. and level for scheme order 1)

Burgers results (MR+MLF sol. and error for scheme order 1)

Burgers results (Error for scheme order 2)

Burgers results (Error for scheme order 3)

Burgers results (Error for scheme order 3)

Burgers results (MR solution and level for scheme order 3)

Burgers results (MR solution and error for scheme order 3)

Burgers results (MR+MLF sol. and level for scheme order 3)

Burgers results (MR+MLF sol. and error for scheme order 3)

Burgers results (MR VS MR+MLF error for scheme order 3)

Burgers 2D results (MR+MLF solution order 1)

Euler 2D results (MR+MLF solution order 3 (OSMP scheme))

Euler 2D results (MR+MLF solution order 3 (OSMP scheme))

Adaptive mesh refinement software

Mesh adaptation

If we look at all the open source software specializing in dynamic mesh adaptation, there are two main families: - patch-based, which is a hierarchical representation of the mesh: layers are placed on top of layers - cell-based, which is a flat representation of the mesh

Each has its advantages and disadvantages: - patch-based has rectangular zones for tiling and optimizing caches. But it generally requires more cells than necessary. - cell-based requires far fewer cells but requires a tree-like structure, which means that you lose the good memory locality you had with patch-based. We use space filling curves such as Morton or Hilbert to find an acceptable locality.

If we look in a little more detail at the functions offered by these software packages, we can group them into 4 main families.

There are two types of data structure, as described above: a list of blocks or a tree.

There are two main adaptation criteria: one is based on a heuristic criterion and depends on the physical problem you are looking at. This could be a gradient, for example. The other is based on a wavelet decomposition that allows you to adapt the mesh without knowing anything about the physical problem, as we have just shown with Haar wavelets and multiresolution.

As you advance in time, you can choose different time steps depending on the resolution of the grid: this is called subcycling. Otherwise, you take the same time step everywhere and, in general, it is the finest grid that guides its value to satisfy a CFL.

Finally, given that the mesh is dynamic, the load balancing must be reviewed regularly during the calculation so that, in a parallel context, all the processes have more or less the same workload. There are two types of method: one based on the space filling curve, where you cut out chunks of the same size following this curve; the other is based on solving a diffusion equation on the workload of the processes.

Open source software

Name	Data structure	Adaptation criteria	Time scheme	Load balancing
AMReX	block	heuristic	global/local	SFC
Dendro	tree	wavelet	global	SFC
Dyablo	tree	heuristic	global	SFC
Peano	tree	-	-	SFC
P4est	tree	-	-	SFC
samurai	interval	heuristic/wavelet	RK/splitting/IMEX time-space/code coupling	SFC/diffusion algorithm

samurai: create a unified framework for testing a whole range
of mesh adaptation methods with the latest generation of numerical schemes.

Here’s an overview of the software we think is interesting to look at today.

Peano and p4est don’t have any adaptation or time-scheme criteria, as they are software programs specializing solely in mesh management.

samurai

Design principles

• Compress the mesh according to the level-wise spatial connectivity along each Cartesian axis.
• Achieve fast look-up for a cell into the structure, especially for parents and neighbors.
• Maximize the memory contiguity of the stored data to allow for caching and vectorization.
• Facilitate inter-level operations which are common in many numerical techniques.

Compression rates

Compression rates

Level	Num. of cells	p4est	samurai (leaves)	samurai (all)	ratio
\(9\)	66379	2.57 Mb	33.68 Kb	121 Kb	21.24
\(10\)	263767	10.25 Mb	66.64 Kb	236.8 Kb	43.28
\(11\)	1051747	40.96 Mb	132.36 Kb	467.24 Kb	87.66
\(12\)	4200559	163.75 Mb	263.6 Kb	927 Kb	176.64
\(13\)	16789627	654.86 Mb	525.9 Kb	1.85 Mb	353.98
\(14\)	67133575	2.61 Gb	1.05 Mb	3.68 Mb	709.24

We can see here that the mesh is very compressed compared to a version using a tree structure.

The total size of the mesh is multiplied by just under 4, as we build different meshes to handle ghost cells and prediction cells.

What’s important to note is that in a parallel context, it doesn’t cost much for each process to have the meshes of its neighboring domains.

Roadmap

Here’s the samurai roadmap for the coming months and probably years.

We now have the possibility of changing the containers in which our fields are stored. We did this so that we could start using Kokkos. So we’d be very interested in working with the people at Cexa on this.

We have received several funding packages that have enabled us to hire two research engineers.

As part of numpex, we also have an engineer position to work on the I/O part of AMR methods in collaboration with Dyablo developers.

Scientific Collaborations

• Lattice Boltzmann methods and multiresolution - Thomas Bellotti (EM2C/CNRS/CS) and Benjamin Graille (LMO/Université Paris-Saclay)
• Plasma discharges and electric propulsion - Alejandro Alvarez-Laguna (LPP/École polytechnique) and Louis Reboul (ONERA) Teddy Pichard and Zoubaïr Tazakkati (CMAP/École polytechnique)
• DNS of lithium-ion batteries based on high-resolution 3D images of porous electrode microstructures - Ali Asad (TotalEnergies) and Laurent François (ONERA)
• Sharp interface method for low Mach two-phase flows - Nicolas Grenier (LISN/Université Paris-Saclay) and Christian Tenaud (EM2C/CNRS/CS)
• Low-Mach reactive flows - Christian Tenaud (EM2C/CNRS/CS)
• Interfacial flow simulation relying on diffuse interface / two-scale modeling - Giuseppe Orlando, Ward Haegeman (CMAP/Ecole polytechnique), Samuel Kokh (CEA/MdlS), Joël Dupays, Clément Le Touze (ONERA), Marica Pelanti (ENSTA/IP Paris), Khaled Saleh (Aix-Marseille Université), Jean-Marc Hérard (EDF)
• High-fidelity simulation of the Hydrogen risk - Luc Lecointre, Pierre-Alexandre Masset, Etienne Studer (CEA), Sergey Koudriakov (CEA) and Christian Tenaud (EM2C/CNRS/CS)

Five projects - NASA Modelling Summer Visit 2025

Five groups from France involving 15 students / post-doctoral fellows / research software engineers / faculties

• Mathematical modeling and simulation of non-equilibrium plasmas for the prediction of electric propulsion and plasma discharges including sheath
• Code coupling : multistep coupling algorithm (Tuesday’s seminar)
• A two-phase flow model with scale separation for the separated-to-disperse phase transition in high-velocity regimes
• A dual splitting/ImEx strategy for multicomponent reacting flows with space/time adaptation and error control
• Simulation of hybrid dense-rarefied flows relying on relaxation schemes and a hierarchy of moment methods

Five projects - NASA Modelling Summer Visit 2025

Five groups from France involving 15 students / post-doctoral fellows / research software engineers / faculties

• Mathematical modeling and simulation of non-equilibrium plasmas for the prediction of electric propulsion and plasma discharges including sheath
• Code coupling : multistep coupling algorithm (Tuesday’s seminar)
• A two-phase flow model with scale separation for the separated-to-disperse phase transition in high-velocity regimes
• A dual splitting/ImEx strategy for multicomponent reacting flows with space/time adaptation and error control
• Simulation of hybrid dense-rarefied flows relying on relaxation schemes and a hierarchy of moment methods

Thank you for your attention

Burgers results (MR solution and level for scheme order 3)

Burgers results (MR solution and error for scheme order 3)

Burgers results (MR+MLF sol. and level for scheme order 3)

Burgers results (MR+MLF sol. and error for scheme order 3)

Burgers results (MR+MLF sol. order 1) small hat

Burgers results (MR+MLF sol. order 1) sinus

\(\epsilon = 1e-2\), \(\underline{\ell} = 2\), \(\bar{\ell} = 12\)