arXiv subjects

Eric Ngondiep

Publications and source records attributed to Eric Ngondiep.

At least 19 recordsLinked to original sources

arXiv2026

A strong second-order two-stage explicit/implicit technique with spectral orthogonal basis Galerkin finite element method for two-dimensional Gray-Scott model

This paper proposes a strong second-order two-step explicit/implicit technique with spectral orthogonal basis Galerkin finite element method for solving a two-dimensional Gray-Scott model subject to appropriate initial and boundary conditions. The constructed approach discretizes at the first stage utilizing a second-order explicit method while a second-order implicit scheme is employed at the second phase. The space derivatives are approximated with the Galerkin finite element formulation combined with a spectral orthogonal basis. With this combination, the errors increased at the first stage are balanced by the errors decreased at the second phase so that the stability is maintained. Furthermore, the use of the spectral orthogonal basis minimizes the space errors. Thus, the new computational approach calculates efficiently numerical solutions and preserves a strong stability and high-order accuracy. The theoretical studies indicate that the proposed strategy is unconditionally stable, temporal second-order accurate and spatial $qth$-order convergent using the $L^{\infty}(0,T;[L^{\infty}(Ω)]^{2})$-norm, where $q$ is an integer greater than or equal $2$. Some numerical examples are performed to confirm the theory and to demonstrate the efficiency of the developed algorithm.

math.NA↗

arXiv2026

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

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.

math.NA↗

arXiv2025

A strong two-stage explicit/implicit approach combined with mixed finite element methods for a three-dimensional nonlinear radiation-conduction model in anisotropic media

This paper develops a strong computational approach to simulate a three-dimensional nonlinear radiation-conduction model in optically thick media, subject to suitable initial and boundary conditions. The space derivatives are approximated by the mixed finite element method ($\mathcal{P}_{p}/\mathcal{P}_{p-1}$), while the interpolation technique is employed in two stages to approximate the time derivative. The proposed strategy is so-called, a strong two-stage explicit/implicit computational technique combined with mixed finite element method. Specifically, the new algorithm should be observed as a predictor-corrector numerical scheme. Additionally, it efficiently treats the time derivative term and provides a necessary requirement on time step for stability. Under this time step limitation, the stability is deeply analyzed whereas the convergence order is numerically computed in the $L^{2}$-norm. The theoretical results suggest that the developed approach is stable and temporal second-order accurate. Some numerical experiments are performed to confirm the theory, to establish that the constructed method is spatial fourth-order convergent and to demonstrate the practical applicability and computational efficiency of the numerical scheme.

math.NA↗

arXiv2025

An unconditionally stable numerical approach for solving a nonlinear distributed delay Sobolev model

This paper proposes an unconditionally stable numerical method for solving a nonlinear Sobolev model with distributed delay. The proposed computational approach approximates the time derivative by interpolation technique whereas the spatial derivatives are approximated using the finite element approximation. This combination is simple and easy to implement. Both stability and error estimates of the constructed method are deeply analyzed in a strong norm which is equivalent to the $H^{1}$-norm. The theoretical results indicate that the constructed approach is unconditionally stable, spatial fourth-order accurate, second-order convergent in time and more efficient than a large class of numerical methods discussed in the literature for solving a general class of delay Sobolev problems. Some numerical examples are carried out to confirm the theory and demonstrate the applicability and validity of the developed technique.

math.NA↗

arXiv2025

An explicit computational approach for a three-dimensional system of nonlinear elastodynamic sine-Gordon problem

This paper proposes an explicit computational method for solving a three-dimensional system of nonlinear elastodynamic sine-Gordon equations subject to appropriate initial and boundary conditions. The time derivative is approximated by interpolation technique whereas the finite element approach is used to approximate the space derivatives. The developed numerical scheme is so-called, high-order explicit computational technique. The new algorithm efficiently treats the time derivative term and provides a suitable time step restriction for stability and convergence. Under this time step limitation, both stability and error estimates of the proposed approach are deeply analyzed using a constructed strong norm. The theoretical studies indicate that the developed approach is temporal second-order convergent and spatially third-order accurate. Some numerical examples are carried out to confirm the theory, to validate the computational efficiency and to demonstrate the practical applicability of the new computational technique.

math.NA↗

arXiv2025

A high-order combined interpolation/finite element technique for evolutionary coupled groundwater-surface water problem

A high-order combined interpolation/finite element technique is developed for solving the coupled groundwater-surface water system that governs flows in karst aquifers. In the proposed high-order scheme we approximate the time derivative with piecewise polynomial interpolation of second-order and use the finite element discretization of piecewise polynomials of degree $d$ and $d+1$, where $d \geq 2$ is an integer, to approximate the space derivatives. The stability together with the error estimates of the constructed technique are established in $L^{\infty}(0,T;\text{\,}L^{2})$-norm. The analysis suggests that the developed computational technique is unconditionally stable, temporal second-order accurate and convergence in space of order $d+1$. Furthermore, the new approach is faster and more efficient than a broad range of numerical methods discussed in the literature for the given initial-boundary value problem. Some examples are carried out to confirm the theoretical results.

math.NA↗

arXiv2025

A combined Lax-Wendroff/interpolation approach with finite element method for a three-dimensional system of tectonic deformation model: application to landslides in Cameroon

This paper develops an efficient computational technique to assess the landslide responses to tectonic deformation and to predict the implications of large bedrocks landslides on the short and long-term development of the disasters. The considered equations represent a three-dimensional system of geological structure deformation subject to suitable initial and boundary conditions. The space derivatives are approximated using the finite element procedure while the approximation in time derivative is obtained using the Lax-Wendroff and interpolation techniques. The new approach is so called a combined Lax-Wendroff/interpolation method with finite element method. The modified Lax-Wendroff/interpolation scheme is employed to efficiently treat the time derivative term and to provide a suitable time step restriction for stability. Under this time step requirement, both stability and error estimates of the new algorithm are deeply analyzed using a constructed strong norm. The theory suggests that the developed computational technique is second-order accurate in time and spatial convergent with order O(h^{p}), where $h$ denotes the space size and p is a positive integer. A wide set of numerical examples are carried out to confirm the theoretical results and to demonstrate the utility and validity of the proposed numerical scheme. An application to landslides observed in west and center regions in Cameroon from October 2019 to November 2024, are discussed.

math.NA↗

arXiv2024

A robust time-split linearized explicit/implicit technique for two-dimensional hydrodynamic model: an application to floods in Cameroon far north region

This paper deals with a time-split explicit/implicit approach for solving a two-dimensional hydrodynamic flow model with appropriate initial and boundary conditions. The time-split technique is employed to upwind the convection term and to treat the friction slope so that the numerical oscillations and stability are well controlled. A suitable time step restriction for stability and convergence accurate of the new algorithm is established using the $L^{\infty}(0,T; L^{2})$-norm. Under a time step requirement, some numerical examples confirm the theoretical studies and suggest that the proposed computational technique is spatial fourth-order accurate and temporal second-order convergent. An application to floods observed in Cameroon far north region is considered and discussed.

math.NA↗

arXiv2023

A Fast Second-Order Explicit Predictor-Corrector Numerical Technique To Investigating And Predicting The Dynamic Of Cytokine Levels And Human Immune Cells Activation In Response To Gram-Positive Bacteria: Staphylococcus Aureus

This paper develops a second-order explicit predictor-corrector numerical approach for solving a mathematical model on the dynamic of cytokine expressions and human immune cell activation in response to the bacterium staphylococcus aureus (S. aureus). The proposed algorithm is at least zero-stable and second-order accurate. Mathematical modeling works that analyze the human body in response to some antigens have predicted concentrations of a broad range of cells and cytokines. This study deals with a coupled cellular-cytokine model which predicts cytokine expressions in response to gram-positive bacteria S. aureus. Tumor necrosis factor alpha, interleukin 6, interleukin 8 and interleukin 10 are included to assess the relationship between cytokine release from macrophages and the concentration of the S. aureus antigen. Ordinary differential equations are used to model cytokine levels while the cellular responses are modeled by partial differential equations. Interactions between both components provide a more robust and complete systems of immune activation. In the numerical simulations, a low concentration of S. aureus is used to measure cellular activation and cytokine expressions. Numerical experiments indicate how the human immune system responds to infections from different pathogens. Furthermore, numerical examples suggest that the new technique is faster and more efficient than a large class of statistical and numerical schemes discussed in the literature for systems of nonlinear equations and can serve as a robust tool for the integration of general systems of initial-boundary value problems.

math.NA↗

arXiv2022

A Robust Three-Level Time Split High-Order Leapfrog/Crank-Nicolson Scheme For Two-Dimensional Sobolev and Regularized Long Wave Equations Arising In Fluid Mechanics

This paper develops a robust three-level time split high-order Leapfrog/Crank-Nicolson technique for solving the two-dimensional unsteady sobolev and regularized long wave equations arising in fluid mechanics. A deep analysis of the stability and error estimates of the proposed approach is considered using the $L^{\infty}(0,T;H^{2})$-norm. Under a suitable time step requirement, the theoretical studies indicate that the constructed numerical scheme is strongly stable (in the sense of $L^{\infty}(0,T;H^{2})$-norm), temporal second-order accurate and convergence of order $O(h^{\frac{8}{3}})$ in space, where $h$ denotes the grid step. This result suggests that the proposed algorithm is less time consuming, fast and more efficient than a broad range of numerical methods widely discussed in the literature for the considered problem. Numerical experiments confirm the theory and demonstrate the efficiency and utility of the three-level time split high-order formulation.

math.NA↗

arXiv2022

A Fast Third-Step Second-Order Explicit Numerical Approach To Investigating and Forecasting The Dynamic of Corruption And Poverty In Cameroon

This paper constructs a third-step second-order numerical approach for solving a mathematical model on the dynamic of corruption and poverty. The stability and error estimates of the proposed technique are analyzed using the $L^{2}$-norm. The developed algorithm is at less zero-stable and second-order accurate. Furthermore, the new method is explicit, fast and more efficient than a large class of numerical schemes applied to nonlinear systems of ordinary differential equations and can serve as a robust tool for integrating general systems of initial-value problems. Some numerical examples confirm the theory and also consider the corruption and poverty in Cameroon.

econ.GN↗

arXiv2022

Unconditional Stability Of A Two-Step Fourth-Order Modified Explicit Euler/Crank-Nicolson Approach For Solving Time-Variable Fractional Mobile-Immobile Advection-Dispersion Equation

This paper considers a two-step fourth-order modified explicit Euler/Crank-Nicolson numerical method for solving the time-variable fractional mobile-immobile advection-dispersion model subjects to suitable initial and boundary conditions. Both stability and error estimates of the new approach are deeply analyzed in the $L^{\infty}(0,T;L^{2})$-norm. The theoretical studies show that the proposed technique is unconditionally stable with convergence of order $O(k+h^{4})$, where $h$ and $k$ are space step and time step, respectively. This result indicate that the two-step fourth-order formulation is more efficient than a broad range of numerical schemes widely studied in the literature for the considered problem. Numerical experiments are performed to verify the unconditional stability and convergence rate of the developed algorithm.

math.NA↗

arXiv2021

A Two-Level Fourth-Order Approach For Time-Fractional Convection-Diffusion-Reaction Equation With Variable Coefficients

This paper develops a two-level fourth-order scheme for solving time-fractional convection-diffusion-reaction equation with variable coefficients subjected to suitable initial and boundary conditions. The basis properties of the new approach are investigated and both stability and error estimates of the proposed numerical scheme are deeply analyzed in the $L^{\infty}(0,T;L^{2})$-norm. The theory indicates that the method is unconditionally stable with convergence of order $O(k^{2-\fracλ{2}}+h^{4})$, where $k$ and $h$ are time step and mesh size, respectively, and $λ\in(0,1)$. This result suggests that the two-level fourth-order technique is more efficient than a large class of numerical techniques widely studied in the literature for the considered problem. Some numerical evidences are provided to verify the unconditional stability and convergence rate of the proposed algorithm.

math.NA↗

arXiv2020

An efficient explicit approach for predicting the Covid-19 spreading with undetected infectious: The case of Cameroon

This paper considers an explicit numerical scheme for solving the mathematical model of the propagation of Covid-19 epidemic with undetected infectious cases. We analyze the stability and convergence rate of the new approach in $L^{\infty}$-norm. The proposed method is less time consuming. Furthermore, the method is stable, at least second-order convergent and can serve as a robust tool for the integration of general systems of ordinary differential equations. A wide set of numerical evidences which consider the case of Cameroon are presented and discussed.

q-bio.PE↗

arXiv2019

A Novel Three-Level Time-Split MacCormack Method for Solving Two-Dimensional Viscous Coupled Burgers Equations

In this paper, we analyze the three-level explicit time-split MacCormack procedure in the numerical solutions of two-dimensional viscous coupled Burgers' equations subject to initial and boundary conditions. The differential operators split the two-dimensional problem into two pieces so that the two-step explicit MacCormack scheme can be easily applied to each subproblem. This reduces the computational cost of the algorithm. For low Reynolds numbers, the proposed method is second order accurate in time and fourth convergent in space, while it is second order convergent in both time and space for high Reynolds numbers problems. This shows the efficiency and effectiveness of the considered method compared to a large set of numerical schemes widely studied in the literature for solving the two-dimensional time dependent nonlinear coupled Burgers' equations. Numerical examples which confirm the theoretical results are presented and discussed.

math.NA↗

arXiv2019

Spectral Distribution in the Eigenvalues Sequence of Products of g-Toeplitz Structures

Starting from the definition of an $n\times n$ $g$-Toeplitz matrix, $T_{n,g}(u)=\left[\widehat{u}_{r-gs}\right]_{r,s=0}^{n-1},$ where $g$ is a given nonnegative parameter, $\{\widehat{u}_{k}\}$ is the sequence of Fourier coefficients of the Lebesgue integrable function $u$ defined over the domain $\mathbb{T}=(-π,π]$, we consider the product of $g$-Toeplitz sequences of matrices, $\{T_{n,g}(f_{1})T_{n,g}(f_{2})\},$ which extends the product of Toeplitz structures, $\{T_{n}(f_{1})T_{n}(f_{2})\},$ in the case where the symbols $f_{1},f_{2}\in L^{\infty}(\mathbb{T}).$ Under suitable assumptions, the spectral distribution in the eigenvalues sequence is completely characterized for the products of $g$-Toeplitz structures. Specifically, for $g\geq2$ our result shows that the sequences $\{T_{n,g}(f_{1})T_{n,g}(f_{2})\}$ are clustered to zero. This extends the well-known result, which concerns the classical case (that is, $g=1$) of products of Toeplitz matrices. Finally, a large set of numerical examples confirming the theoretic analysis is presented and discussed.

math.NA↗

arXiv2019

A Time-Split MacCormack Scheme for Two-Dimensional Nonlinear Reaction-Diffusion Equations

A three-level explicit time-split MacCormack scheme is proposed for solving the two-dimensional nonlinear reaction-diffusion equations. The computational cost is reduced thank to the splitting and the explicit MacCormack scheme. Under the well known condition of Courant-Friedrich-Lewy (CFL) for stability of explicit numerical schemes applied to linear parabolic partial differential equations, we prove the stability and convergence of the method in $L^{\infty}(0,T;L^{2})$-norm. A wide set of numerical evidences which provide the convergence rate of the new algorithm are presented and critically discussed.

math.NA↗

arXiv2019

Error Estimate of MacCormack Rapid Solver Method for 2D Incompressible Navier-Stokes Problems

The error estimates and convergence rate of a two-level MacCormack rapid solver method for solving a two-dimensional incompressible Navier-Stokes equations are analyzed. This represents a continuation of the work on the stability analysis of the method. The theoretical result suggests that the rapid solver method is both convergent and second order accurate with respect to time step $Δt.$ A wide set of numerical evidences confirm this theoretical analysis.

math.NA↗