A dual splitting/ImEx strategy for multicomponent reacting flows with space /time adaptation and error control

Josselin Massot, Laurent Series, Christian Tenaud, Loïc Gouarin, Pierre Matalon, Marc Massot

Mesh adaptation and error control

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: \(\Delta x_{\ell}:=2^{-\ell}=2^{\Delta \ell} \Delta x\).

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].

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

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

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

Numerical Analysis

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

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].

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

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

Burgers results

Burgers results (Error for scheme order 1)

Burgers results (MR solution for scheme order 1)

Burgers results (MR+MLF solution for scheme order 1)

Burgers results (Error for scheme order 1)

Burgers results (MR solution for scheme order 1)

Burgers results (MR+MLF solution for scheme order 1)

Burgers results (MR+MLF solution order 1) sinus

A dual ImEx/Splitting strategy for stiff PDEs

A strategy has been designed (PhD M. Duarte) relying on time-adaptive operator splitting with dynamically adapted mesh (multiresolution) and error control:
- optimal computational cost and parallelization properties (large splitting time steps)
We aim at resolving stiff PDEs with samurai and ponio libraries with the same computation favorable properties:
- local implicitation of the source term
- explicit diffusion integration without von Neumann stability limit (ROCK)
- high-order in space and time integration of convection, including shocks
- strong acceleration through adaptation in space and time
Stumbling block: what to do when reaction and diffusion coupled at smallest time scale (complex chem. - ignition)

An ImEx strategy for stiff reaction-diffusion

Belousov-Zhabotinsky (very stiff source - 3 eq) \[ \left\{ \begin{aligned} \partial_t a - D_a \, \Delta a &= \frac{1}{\mu} ( -qa - ab + fc) \\ \partial_t b - D_b \, \Delta b &= \frac{1}{\varepsilon} ( qa - ab + b\,(1-b)) \\ \partial_t c - D_c \, \Delta c &= b - c \end{aligned} \right. \]

Error to the reference quasi-exact solution is second order in time but not of the same origin (splitting error vs. IMEX error) - but still error control
Larger time step can be taken with IMEX while keeping a proper solution (no disastrous splitting errors - wrong wave speed)
When optimal large splitting time step is taken, IMEX as efficient as splitting, whereas it is advantageous for smaller time steps as well as larger time steps
No boundary condition problems
Same computational good properties

Simulation with ponio / samurai

\(\epsilon = 1e-3\), \(\underline{\ell} = 2\), \(\bar{\ell} = 10\)

Conclusion

Analysis of mesh adaptation completed for convection - extension to reaction and diffusion
- Link with AMR important
ImEx strategy implemented in ponio - coupled to samurai
- Implicit source term / explicit diffusion (no von Neumann stability limit - ROCK)
Extension to convection on the way - required fine error analysis
- high-order in space and time integration of convection, including shocks (OSMP)