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An efficient predictor-corrector approach with orthogonal spline collocation finite element technique for FitzHugh-Nagumo problem

Eric Ngondiep

Abstract

This paper constructs a predictor-corrector technique with orthogonal spline collocation finite element method for simulating a FitzHugh-Nagumo system subject to suitable initial and boundary conditions. The developed computational technique approximates the exact solution in time using variable time steps at the predictor phase and a constant time step at the corrector stage while the orthogonal spline collocation finite element method is employed in the space discretization. The new algorithm presents several advantages: (i) the errors increased at the predictor phase are balanced by the ones decreased at the corrector phase so that the stability is preserved, (ii) the variable time steps at the predictor stage overcome the numerical oscillations, (iii) the spatial errors are minimized due to the use of collocation nodes, and (iv) the linearization of the nonlinear term reduces the required operations at the corrector stage. As a result, the new computational technique computes efficiently both predicted and corrected solutions and preserves a strong stability and high-order accuracy even in the presence of singularities. The theoretical results suggest that the constructed approach is unconditionally stable, spatial $mth$-order accurate and temporal second-order convergent in the $L^{\infty}(0,T;[H^{m}(Ω)]^{2})$-norm. Some numerical experiments are carried out to confirm the theoretical analysis and to demonstrate the applicability and performance of the proposed strategy.

An efficient predictor-corrector approach with orthogonal spline collocation finite element technique for FitzHugh-Nagumo problem

Abstract

This paper constructs a predictor-corrector technique with orthogonal spline collocation finite element method for simulating a FitzHugh-Nagumo system subject to suitable initial and boundary conditions. The developed computational technique approximates the exact solution in time using variable time steps at the predictor phase and a constant time step at the corrector stage while the orthogonal spline collocation finite element method is employed in the space discretization. The new algorithm presents several advantages: (i) the errors increased at the predictor phase are balanced by the ones decreased at the corrector phase so that the stability is preserved, (ii) the variable time steps at the predictor stage overcome the numerical oscillations, (iii) the spatial errors are minimized due to the use of collocation nodes, and (iv) the linearization of the nonlinear term reduces the required operations at the corrector stage. As a result, the new computational technique computes efficiently both predicted and corrected solutions and preserves a strong stability and high-order accuracy even in the presence of singularities. The theoretical results suggest that the constructed approach is unconditionally stable, spatial -order accurate and temporal second-order convergent in the -norm. Some numerical experiments are carried out to confirm the theoretical analysis and to demonstrate the applicability and performance of the proposed strategy.
Paper Structure (5 sections, 7 theorems, 175 equations, 3 figures)

This paper contains 5 sections, 7 theorems, 175 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Construction of the new computational technique
  3. Stability analysis and error estimates
  4. Numerical experiments
  5. General conclusions and future investigations

Key Result

Lemma 3.1

For every $Q_{i}\in\mathcal{F}_{h}$, and any $U\in[H^{1}(\Omega)]^{2}$, the orthogonal projections $\mathbb{P}_{h}$ and $P_{h}$ defined in equation $()$, satisfy where $\bar{\nabla}_{r}=[\nabla_{r},\nabla_{r}]$, denotes the restriction of the operator $\bar{\nabla}$ (defined by equation $()$) on $Q_{i}$, for $i=1,2,...,M$, that is, for all $U\in[H^{1}(\Omega)]^{2}$, $\bar{\nabla}U|_{Q_{i}}=\bar{\

Figures (3)

  • Figure 1: exact solution, approximate solution and error corresponding to Example 1
  • Figure 2: exact solution, approximate solution and error associated with Example 2
  • Figure 3: numerical solution associated with Example 3

Theorems & Definitions (14)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Corollary 3.1
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • ...and 4 more