Conservative integrators for piecewise smooth systems with transversal dynamics

Anil N. Hirani; Andy T. S. Wan; Nikolas Wojtalewicz

doi:10.3934/jcd.2023009

Article Contents

2024, Volume 11, Issue 2: 135-152. Doi: 10.3934/jcd.2023009

This issue Previous Article Next Article Towards optimal space-time discretization for reachable sets of nonlinear control systems

Conservative integrators for piecewise smooth systems with transversal dynamics

1.
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL, USA
2.
Department of Applied Mathematics, University of California Merced, Merced, CA, USA
3.
Department of Mathematics and Statistics, University of Northern British Columbia, Prince George, BC, Canada

^*Corresponding author: Nikolas Wojtalewicz
^*Corresponding author: Nikolas Wojtalewicz

Received: July 2023

Revised: October 2023

Early access: November 2023

Published: April 2024

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Abstract

We introduce conservative integrators for long term integration of piecewise smooth systems with transversal dynamics and piecewise smooth conserved quantities. In essence, for a piecewise dynamical system with piecewise defined conserved quantities where its trajectories cross transversally to its interface, we combine Mannshardt's transition scheme and the Discrete Multiplier Method to obtain conservative integrators capable of preserving conserved quantities up to machine precision and accuracy order. We prove that the order of accuracy of the conservative integrators is preserved after crossing the interface in the case of codimension one number of conserved quantities. Numerical examples in two and three dimensions illustrate the preservation of accuracy order across the interface for cubic and logarithmic type conserved quantities. We observed that conservative transition schemes can prevent spurious transitions from occurring, even in the case when there are fewer conserved quantities.

Keywords:

Mathematics Subject Classification: Primary: 65L05, 65L12, 65L20, 65L70, 65P10, 37M05, 37M15.

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Figure 1. (A) PWS trajectories for the elliptic curve example with the interface being a circle of radius 1 centered at the origin. (B) Convergence of the error between the exact transition time $ t^* $ and the approximate transition time $ \hat{t} $ for the PWS elliptic curve system

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Figure 2. Comparison of error in the solution of the PWS elliptic curve system with different perturbation added to $ \hat{t} $ each time the interface is crossed

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Figure 3. Comparison of the error in conserved quantity versus time between our DMM transition scheme and the RK2 transition scheme for the PWS elliptic curve system. An initial condition of $ [-1,-1] $ and a step size $ 10^{-3} $ was used for both methods

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Figure 4. (A) An initial condition of $ [0.612, 0.0137, 0.629] $ was used with a final time $ T = 700 $ and step size $ \tau = 0.07 $. Here DMM transition scheme agrees with the RK4 transition scheme, but the RK2 transition scheme has drifted far away due to spurious transitions. (B) Convergence of the error between the exact transition time $ t^* $ and the approximate transition time $ \hat{t} $ for the PWS three species Lotka-Volterra system

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Figure 5. Comparison of error in the solution of the PWS three species Lotka-Volterra system with different perturbation added to $ \hat{t} $ each time the interface is crossed

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Figure 6. Comparison of the error in conserved quantity versus time, as computed by RK2 (top), RK4 (middle) and our DMM transition scheme (bottom) for the PWS three species Lotka-Voltera system

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Figure 7. The sign of the switching function $ g $ for the three methods are plotted versus time, corresponding tok__ge Figure 6. In this case, the solution computed using RK2 makes spurious transitions (denoted by red "x") while the RK4 and DMM solutions do not

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Figure 8. Comparison of error in conserved quantity versus time, as computed by RK2 (top), RK4 (middle) and our DMM transition scheme (bottom) for the PWS three species Lotka-Voltera system when the solution makes several transitions

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Figure 9. The sign of the switching function $ g $ for the three methods are plotted versus time, corresponding to Figure 8. All three methods make initial transitions and then RK2 makes further spurious transitions (denoted by red "x")

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Table .

Algorithm 1: Mannshardt's transition scheme (uniform time steps)
Given $ \tau,t_0,\mathit{\boldsymbol{x}}_0 $.
for $ k=0,1,2,\dots $ do

end

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