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Modified Patankar Linear Multistep methods for production-destruction systems

Giuseppe Izzo, Eleonora Messina, Mario Pezzella, Antonia Vecchio

TL;DR

This work develops Modified Patankar Linear Multistep (MPLM) methods for Production-Destruction Systems, extending the Patankar trick to linear multistep schemes that preserve positivity and the linear invariants of the continuous system for any step size. Central to the approach is the Patankar-Weight Denominator (PWD) and a $oldsymbol{\sigma}$-embedding technique that yields arbitrarily high order convergence by recursively computing PWDs from lower-order MPLM steps. The authors prove sufficient and necessary conditions for order-$p$ convergence and establish global convergence under regularity assumptions, accompanied by a practical algorithm to obtain PWDs. Numerical experiments on linear, nonlinear, Brusselator, epidemiological, and diffusion problems corroborate the theory, showing competitive or superior accuracy-for-cost relative to existing high-order Patankar schemes, and demonstrating preservation of positivity and mass conservation in challenging scenarios.

Abstract

Modified Patankar schemes are linearly implicit time integration methods designed to be unconditionally positive and conservative. In the present work we extend the Patankar-type approach to linear multistep methods and prove that the resulting discretizations retain, with no restrictions on the step size, the positivity of the solution and the linear invariant of the continuous-time system. Moreover, we provide results on arbitrarily high order of convergence and we introduce an embedding technique for the Patankar weight denominators to achieve it.

Modified Patankar Linear Multistep methods for production-destruction systems

TL;DR

This work develops Modified Patankar Linear Multistep (MPLM) methods for Production-Destruction Systems, extending the Patankar trick to linear multistep schemes that preserve positivity and the linear invariants of the continuous system for any step size. Central to the approach is the Patankar-Weight Denominator (PWD) and a -embedding technique that yields arbitrarily high order convergence by recursively computing PWDs from lower-order MPLM steps. The authors prove sufficient and necessary conditions for order- convergence and establish global convergence under regularity assumptions, accompanied by a practical algorithm to obtain PWDs. Numerical experiments on linear, nonlinear, Brusselator, epidemiological, and diffusion problems corroborate the theory, showing competitive or superior accuracy-for-cost relative to existing high-order Patankar schemes, and demonstrating preservation of positivity and mass conservation in challenging scenarios.

Abstract

Modified Patankar schemes are linearly implicit time integration methods designed to be unconditionally positive and conservative. In the present work we extend the Patankar-type approach to linear multistep methods and prove that the resulting discretizations retain, with no restrictions on the step size, the positivity of the solution and the linear invariant of the continuous-time system. Moreover, we provide results on arbitrarily high order of convergence and we introduce an embedding technique for the Patankar weight denominators to achieve it.
Paper Structure (13 sections, 9 theorems, 76 equations, 22 figures, 6 tables)

This paper contains 13 sections, 9 theorems, 76 equations, 22 figures, 6 tables.

Table of Contents

  1. Introduction
  2. The Modified Patankar Linear Multistep scheme
  3. Error analysis and convergence
  4. The $\sigma$-embedding technique
  5. Numerical Experiments
  6. Test 1: linear test
  7. Test 2: nonlinear test
  8. Test 3: Brusselator test
  9. Test 4: SACEIRQD COVID-19 model
  10. Test 5: spatially heterogeneous diffusion equation
  11. Spatial semi-discretization and conservative PDS
  12. Simulation results
  13. Conclusions and perspectives

Key Result

Lemma 2.2

Let $\{\bm{y}^n\}_{n\geq k}$ be the approximation of the solution to eq:sistema_compatto computed by eq:Compact_Expression. Suppose that eq: pos_cont and cond:b hold true and assume that Then, for all $h >0$ and $n\geq0,$$\bm{y}^n>0.$

Figures (22)

  • Figure 1: Numerical solution of \ref{['Test1']} computed by MPLM-$10(6)$ with $h=2^{-5}$ .
  • Figure 2: Experimental order for MPE, MPLM-$2(2),$ MPLM-$4(3),$ MPLM-$5(4)$ and MPRK3 applied to \ref{['Test1']}.
  • Figure 3: Experimental order for MPLM-$7(5),$ MPLM-$10(6)$ and MPRK3 applied to \ref{['Test1']}.
  • Figure 4: Work Precision Diagram: Error versus CPU time for the different methods applied to \ref{['Test1']}. $h=T/2^{5+m}$, $m=1,\ldots,7.$
  • Figure 5: Work Precision Diagram: Mean Error versus CPU time for the different methods applied to \ref{['Test1']}. $h=T/2^{5+m}$, $m=1,\ldots,7.$
  • ...and 17 more figures

Theorems & Definitions (19)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • Lemma 3.3
  • proof
  • ...and 9 more