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Splitting Schemes for ODEs with Goal-Oriented Error Estimation

Erik Weyl, Andreas Bartel, Manuel Schaller

TL;DR

This work addresses solving time-dependent ODEs arising from coupled subsystems with distinct time scales by fusing dynamic iteration (DI) with finite elements in time and a goal-oriented error framework. The authors develop a hybrid a-priori/a-posteriori estimator that blends DI error bounds with a dual weighted residual (DWR) error estimator to balance iteration and discretization errors in a specified QoI, $J(U)$. A holistic stacked variational formulation for the DI process is derived, enabling local goal-oriented estimators and an algorithm that drives adaptive, component-wise multiadaptive time discretization while providing a stopping criterion for the iteration. Numerical experiments show that goal-oriented refinement often outperforms uniform refinement, particularly in multi-rate scenarios, though the DWR-based bounds can be conservative for higher-order schemes like Crank-Nicolson.

Abstract

We present a hybrid a-priori/a-posteriori goal oriented error estimator for a combination of dynamic iteration-based solution of ordinary differential equations discretized by finite elements. Our novel error estimator combines estimates from classical dynamic iteration methods, usually used to enable splitting-based distributed simulation, and from the dual weighted residual method to be able to evaluate and balance both, the dynamic iteration error and the discretization error in desired quantities of interest. The obtained error estimators are used to conduct refinements of the computational mesh and as a stopping criterion for the dynamic iteration. In particular, we allow for an adaptive and flexible discretization of the time domain, where variables can be discretized differently to match both goal and solution requirements, e.g. in view of multiple time scales. We endow the scheme with efficient solvers from numerical linear algebra to ensure its applicability to complex problems. Numerical experiments compare the adaptive approach to a uniform refinement.

Splitting Schemes for ODEs with Goal-Oriented Error Estimation

TL;DR

This work addresses solving time-dependent ODEs arising from coupled subsystems with distinct time scales by fusing dynamic iteration (DI) with finite elements in time and a goal-oriented error framework. The authors develop a hybrid a-priori/a-posteriori estimator that blends DI error bounds with a dual weighted residual (DWR) error estimator to balance iteration and discretization errors in a specified QoI, . A holistic stacked variational formulation for the DI process is derived, enabling local goal-oriented estimators and an algorithm that drives adaptive, component-wise multiadaptive time discretization while providing a stopping criterion for the iteration. Numerical experiments show that goal-oriented refinement often outperforms uniform refinement, particularly in multi-rate scenarios, though the DWR-based bounds can be conservative for higher-order schemes like Crank-Nicolson.

Abstract

We present a hybrid a-priori/a-posteriori goal oriented error estimator for a combination of dynamic iteration-based solution of ordinary differential equations discretized by finite elements. Our novel error estimator combines estimates from classical dynamic iteration methods, usually used to enable splitting-based distributed simulation, and from the dual weighted residual method to be able to evaluate and balance both, the dynamic iteration error and the discretization error in desired quantities of interest. The obtained error estimators are used to conduct refinements of the computational mesh and as a stopping criterion for the dynamic iteration. In particular, we allow for an adaptive and flexible discretization of the time domain, where variables can be discretized differently to match both goal and solution requirements, e.g. in view of multiple time scales. We endow the scheme with efficient solvers from numerical linear algebra to ensure its applicability to complex problems. Numerical experiments compare the adaptive approach to a uniform refinement.
Paper Structure (15 sections, 3 theorems, 60 equations, 5 figures, 1 algorithm)

This paper contains 15 sections, 3 theorems, 60 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Global view on dynamic iteration
  3. Construction of splitting schemes
  4. A holistic variational formulation for dynamic iteration
  5. Dual weighted residual method
  6. Decoupling
  7. Discretization
  8. Multiadaptive discretization schemes
  9. Solution of the discretized problem
  10. Goal oriented mesh refinement
  11. Local goal oriented discretization error estimators
  12. Algorithm
  13. Numerical Experiments
  14. Conclusion
  15. Proof of DWR Error Estimator

Key Result

Proposition 1

We consider problem eq:1010. Given a splitting matrix $S$ with split model eq:2020, then the error $E_{K}$eq:splitting-error after $K$ iterations leads to an error in the QoI eq:1020$J \left( E_{K} \right)$, which is bounded as follows with $L_1 = \mu_{\max} \left( \frac{\hat{B} + \hat{B}^T}{2} \right)$, $L_2 = \left\| \check{B} \right\|$.

Figures (5)

  • Figure 1: Test Case \ref{['exp:6110']}. Top left: right-hand side. Bottom left: analytical solution. The "+" markers denote the time points of interest which we try to approximate. The larger marker for $u_2$ at $t = 3$ represents the double weight there. Right: Error plots. The $x$--axis displays the total number of time steps made over all components; the $y$--axis shows the error (solid) in $J$ or the estimator (dashed). The goal oriented explicit Euler scheme (blue, up marker) performs better than its uniform counterpart (orange, right marker) in both cases, the error is bounded from above by the estimator. Lower errors can be achieved by using the higher order Crank--Nicolson scheme; with uniform refinement (red, left marker), this yields a very consistent error reduction as $N$ grows while goal oriented refinement (green, down marker) leads to the best results of the examined methods. However, for the Crank--Nicolson schemes, the error is underestimated.
  • Figure 2: Test Case \ref{['exp:6210']}. Top left: right--hand side with slow components $y_1$, $y_2$ and faster components $y_3$, $y_4$. Bottom left: analytical solution with slow components $u_1$, $u_2$ and faster components $u_3$, $u_4$. The "+" also denote the points of interest; We measure $u_2$ at $t = 0.5$ and at $t = 2.5$, we take $u_3$. Right: Convergence results. For the explicit Euler scheme, we see that the goal oriented refinement (blue, down marker) outperforms uniform (orange, right marker) which does not work well in this case. The uniform Crank--Nicolson scheme (red, left marker) reliably reduces the error but also underestimates it (dashed line below solid line). Finally, the goal oriented Crank--Nicolson scheme (green, down marker) works very well here although the error does not monotonically decrease as $l$ is incremented. While the estimator does at times fall below the actual error, the effect is less prominent in this test case due to the high accuracy of goal oriented Crank--Nicolson.
  • Figure 3: Test Case \ref{['exp:6210']} and final approximations for goal oriented grid refinement. The time steps are marked with an $X$ each. Left: explicit Euler scheme. Right: Crank--Nicolson scheme. The refinement adapts to the goal functional, here represented by a vertical line, as described above. It is also notable that the Crank--Nicolson scheme while having the same pattern of refinement as the explicit Euler scheme distributes the time steps slightly more evenly.
  • Figure 4: Test Case \ref{['exp:6310']}. Top left: right-hand side. Bottom left: analytical solution with points of interest marked by "+". Right: convergence plots for the goal oriented dynamic iteration refinements. The goal oriented approach offers less of an advantage here than in the previous tests. In particular, for the explicit Euler scheme, the goal oriented (blue, up marker) refinement performs worse than the uniform (orange, right marker) one. For the Crank--Nicolson scheme, goal oriented (green, down marker) still works better than uniform (red, left marker) but the difference is smaller than in other cases. The stronger coupling compared to Test Case \ref{['exp:6110']} makes it more difficult to exploit the different frequencies and points of interest of the components.
  • Figure 5: Test Case \ref{['exp:6310']}. Iteration steps ($K$) per refinement level for the goal oriented schemes. Explicit Euler (blue, up marker) scheme increases the number of iterations for finer meshes while the Crank--Nicolson scheme (green, down marker) does not shown a clear pattern.

Theorems & Definitions (10)

  • Proposition 1
  • proof
  • Remark 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Remark 2
  • Remark 3
  • Remark 4