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Numerical solutions of random mean square Fisher-KPP models with advection

M. -C. Casabán, R. Company, L. Jódar

TL;DR

This work develops a numerically stable framework for random mean-square Fisher-KPP models with advection by combining spatial semidiscretization with a random exponential time differencing scheme. It proves mean-square stability and positivity under appropriate mesh restrictions, using Metzler matrices and logarithmic norms. A random Lp calculus underpins the analysis, transforming nonlinear random ODEs into computable integral forms. Numerical experiments on a heterogeneous test problem demonstrate that the method accurately reproduces the mean and variance of the solution, validating the approach for uncertainty quantification in reaction-diffusion-advection systems.

Abstract

This paper deals with the construction of numerical stable solutions of random mean square Fisher-KPP models with advection. The construction of the numerical scheme is performed in two stages. Firstly, a semidiscretization technique transforms the original continuous problem into a nonlinear unhomogeneous system of random differential equations. Then, by extending to the random framework the ideas of the exponential time differencing method, a full vector discretization of the problem addresses to a random vector difference scheme. A sample approach of the random vector difference scheme, the use of properties of Metzler matrices and the logarithmic norm allow the proof of stability of the numerical solutions in the mean square sense. In spite of the computational complexity the results are illustrated by comparing the results with a test problem where the exact solution is known.

Numerical solutions of random mean square Fisher-KPP models with advection

TL;DR

This work develops a numerically stable framework for random mean-square Fisher-KPP models with advection by combining spatial semidiscretization with a random exponential time differencing scheme. It proves mean-square stability and positivity under appropriate mesh restrictions, using Metzler matrices and logarithmic norms. A random Lp calculus underpins the analysis, transforming nonlinear random ODEs into computable integral forms. Numerical experiments on a heterogeneous test problem demonstrate that the method accurately reproduces the mean and variance of the solution, validating the approach for uncertainty quantification in reaction-diffusion-advection systems.

Abstract

This paper deals with the construction of numerical stable solutions of random mean square Fisher-KPP models with advection. The construction of the numerical scheme is performed in two stages. Firstly, a semidiscretization technique transforms the original continuous problem into a nonlinear unhomogeneous system of random differential equations. Then, by extending to the random framework the ideas of the exponential time differencing method, a full vector discretization of the problem addresses to a random vector difference scheme. A sample approach of the random vector difference scheme, the use of properties of Metzler matrices and the logarithmic norm allow the proof of stability of the numerical solutions in the mean square sense. In spite of the computational complexity the results are illustrated by comparing the results with a test problem where the exact solution is known.
Paper Structure (8 sections, 8 theorems, 103 equations, 3 figures, 1 algorithm)

This paper contains 8 sections, 8 theorems, 103 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and new results on $L_{p}$-random matrix calculus
  3. Random linear non homogeneous matrix differential systems
  4. Discretization and random numerical scheme construction
  5. Properties of the random numerical scheme
  6. Numerical example
  7. Conclusions
  8. Acknowledgements

Key Result

Proposition 1

(AMM_Xelo_JC_Lucas2016) Let $X=(x_{i,k})\in{L^{m \times q}_{2p}(\Omega)}$ and $Y=(y_{k,j})\in{L^{q \times n}_{2p}(\Omega)}$. Then

Figures (3)

  • Figure 1: Plot (a): Surface of the expectation, $\mathbb{E}[u(x_i,t^n)]$. Plot (b): Surface of the standard deviation, $\sqrt{\mathrm{Var}[u(x_i,t^n)]}$. Both statistical moment functions correspond to the exact solution s.p. \ref{['exactSolEj']} of the problem \ref{['eqEj']}--\ref{['CC2eqEj']}, on the domain $(x_i=ih,t^n=nk)\in{[0,\,1]\times [0, \, T=0.01]}$ for $0 \leq i \leq N=10$ and $0 \leq n \leq N_T=5$.
  • Figure 2: Plot (a): Comparative graphics between the exact values of expectation of \ref{['exactSolEj']}, $\mathbb{E}[u(x_i,T)]$, and the approximated expectation, $\mathbb{E}[u_i^{N_T}]$, using the random numerical scheme \ref{['EsquemaRandom']}. Plot (b): Absolute error of the expectations represented in plot (a). In both graphics we have considered $T=0.01$, i.e $N_T=5$ for $k=0.002$, and $h=0.1$.
  • Figure 3: Plot (a): Comparative graphics between the exact values of standard deviation of \ref{['exactSolEj']}, $\sqrt{\mathrm{Var}[u(x_i,T)]}$, and the approximated standard deviation, $\sqrt{\mathrm{Var}\left[u_i^{N_T}\right]}$, using the random numerical scheme \ref{['EsquemaRandom']}. Plot (b): Absolute error of the standard deviations represented in plot (a). In both graphics we have considered $T=0.01$, i.e $N_T=5$ for $k=0.002$, and $h=0.1$.

Theorems & Definitions (15)

  • Proposition 1
  • Definition 1
  • Lemma 1
  • Proof 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Remark 1
  • Remark 2
  • Lemma 2
  • ...and 5 more