Eric Ngondiep

Eric Ngondiep

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.

Eric Ngondiep, Alqahtani T. Rubayyi, Jean C. Ntonga

In the last decades, more or less complex physically-based hydrological models, have been developed to solve the shallow water equations or their approximations using various numerical methods. The MacCormack method was developed for simulating overland flow with spatially variable infiltration and microtopography using the hydrodynamic flow equations. The basic MacCormack scheme is enhanced when it uses the method of fractional steps to treat the friction slope or a stiff source term and to upwind the convection term in order to control the numerical oscillations and stability. In this paper we describe, the MacCormack scheme for 1D complete shallow water equations with source terms, analyze the stability condition of the method and we provide the convergence rate of the algorithm. This work improves some well known results deeply studied in the literature which concern the Saint-Venant problem and it represents an extension of the time dependent shallow water equations without source terms. The numerical evidences consider the rate of convergence of the method and compares the numerical solution respect to the analytical one.

Eric Ngondiep

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.

Eric Ngondiep

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.

Eric Ngondiep

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.

Eric Ngondiep, Stefano Serra-Capizzano, Debora Sesana

For a given nonnegative integer alpha, a matrix A_{n} of size n is called alpha-Toeplitz if its entries obey the rule A_{n}=[a_{r-alpha*s}]_{r,s=0}^{n-1}. Analogously, a matrix A_{n} again of size n is called alpha-circulant if A_{n}= [a_{(r-alpha*s)mod n}]_{r,s=0}^{n-1}. Such kind of matrices arises in wavelet analysis, subdivision algorithms and more generally when dealing with multigrid/multilevel methods for structured matrices and approximations of boundary value problems. In this paper we study the singular values of alpha-circulants and we provide an asymptotic analysis of the distribution results for the singular values of alpha-Toeplitz sequences in the case where {a_{k}} can be interpreted as the sequence of Fourier coeffcients of an integrable function f over the domain (-pi;pi). Some generalizations to the block, multilevel case, amounting to choose f multivariate and matrix valued, are briefly considered.