Narrow Results

The stochastic Newell–Whitehead–Segel in $(2+1)$ dimensions is under consideration. It represents the population density or dimensionless temperature and it discusses how stripes appear in temporal and spatial dimensional systems. The Newell–Whitehead–Segel equation (NWSE) has applications in different areas such as ecology, chemical, mechanical, biology and bio-engineering. The important thing is if we see the problem in the two-dimensional (2D) manifold, then the whole 3D picture can be included in the model. The 3D space is embedded compactly in the 2D manifolds. So, 2D problems for the Newell–White–Segel equation are very important because they consider the one, two and three dimensions in it. The numerical solutions of the underlying model have been extracted successfully by two schemes, namely stochastic forward Euler (SFE) and the proposed stochastic nonstandard finite difference (SNSFD) schemes. The existence of the solution is guaranteed by using the contraction mapping principle and Schauder’s fixed-point theorem. The consistency of each scheme is proved in the mean square sense. The stability of the schemes is shown by using von Neumann criteria. The SFE scheme is conditionally stable and the SNSFD scheme is unconditionally stable. The efficacy of the proposed methods is depicted through the simulations. The 2D and 3D graphs are plotted for various values of the parameters.

This investigation aims to examine the resolution of the generalized nonlinear time fractional Harry Dym $(𝔾\mathbb{N}𝕋𝔽ℍ𝔻)$ equation through a combined application of analytical and numerical methodologies. The primary objective is to scrutinize the equation’s behavior and present proficient methodologies for its resolution. The Khater II analytical technique and numerical frameworks, specifically the Cubic-B-spline, Quantic-B-spline, and Septic-B-spline schemes, are proposed for this purpose. The outcomes of this inquiry yield substantial revelations regarding the characteristics of $𝔾\mathbb{N}𝕋𝔽ℍ𝔻$ equation. Both the analytical approach and numerical schemes demonstrate their efficacy in yielding precise solutions. These findings carry noteworthy implications across various disciplines, encompassing physics, mathematics, and engineering. The investigated model appears in plasma physics research on soliton theory and nonlinear waves. Soliton waves, which keep their shape and velocity while propagating, are found in plasma physics and other domains. In plasma environments, the Harry Dym equation describes these solitons and their behavior. Solitons are essential for understanding plasma dynamics, including nonlinear waves and structures. They help comprehend plasma dynamics, wave interactions, and other nonlinear processes. The principal deductions drawn from this study underscore the effectiveness and viability of the proposed techniques in resolving the $𝔾\mathbb{N}𝕋𝔽ℍ𝔻$ equation. This research introduces innovative contributions in terms of insights and methodologies pertinent to analogous nonlinear fractional equations. Its scope encompasses nonlinear dynamics, fractional calculus, and numerical analysis.

In this paper, we present a multi-fluid ionization model. We prove that this stationary, mono-dimensional model has a maximal solution which is not global at variance with the mono-species case and we present a numerical method for solving this highly singular system of ordinary differential equations. Numerical results are compared with those obtained for other models.

We analyze a hyperbolic system of conservation laws in dimension one, which is a drastic simplification of a multi-phase or multi-velocity fluid model. The physical domain of hyperbolicity is bounded, which is a characteristic of multi-phase models. Our main result is the stability of the domain of hyperbolicity. Due to the degeneracy of the model on the boundary of the hyperbolicity domain, rarefaction waves are not unique. We also propose a numerical scheme for approximate resolution of the model and prove the stability of this scheme.

This work aims at providing a mathematical and numerical framework for the analysis on the effects of pulsed electric fields on the physical media that have a heterogeneous permittivity and a heterogeneous conductivity. Well-posedness of the model interface problem and the regularity of its solutions are established. A fully discrete finite element scheme is proposed for the numerical approximation of the potential distribution as a function of time and space simultaneously for an arbitrary-shaped pulse, and it is demonstrated to enjoy the optimal convergence order in both space and time. The new results and numerical scheme have potential applications in the fields of electromagnetism, medicine, food sciences, and biotechnology.

We study stability, dispersion and dissipation properties of four numerical schemes (Itera-tive Crank–Nicolson, 3rd and 4th order Runge–Kutta and Courant–Fredrichs–Levy Nonlinear). By use of a Von Neumann analysis we study the schemes applied to a scalar linear wave equation as well as a scalar nonlinear wave equation with a type of nonlinearity present in GR-equations. Numerical testing is done to verify analytic results. We find that the method of lines (MOL) schemes are the most dispersive and dissipative schemes. The Courant–Fredrichs–Levy Nonlinear (CFLN) scheme is most accurate and least dispersive and dissipative, but the absence of dissipation at Nyquist frequency, if fact, puts it at a disadvantage in numerical simulation. Overall, the 4th order Runge–Kutta scheme, which has the least amount of dissipation among the MOL schemes, seems to be the most suitable compromise between the overall accuracy and damping at short wavelengths.

We formulate second order finite difference schemes based on Padé rational approximations for the numerical integration of nonlinear age-dependent population models. The schemes are completely analysed and some numerical experiments are also reported in order to show numerically their accuracy.

We prove that suitable weak solutions of 3D Navier–Stokes equations in bounded domains can be constructed by a particular type of artificial compressibility approximation.

We consider the strong solution of the 2D Navier–Stokes equations in a torus subject to an additive noise. We implement a fully implicit time numerical scheme and a finite element method in space. We prove that the space-time rate of convergence is the “optimal” one, namely, $\eta \in [0,1/2)$ in time and 1 in space. Let us mention that the coefficient $\eta$ is equal to the time regularity of the solution with values in ${𝕃}^2$. Our method relies on the existence of finite exponential moments for both the solution and its time approximation. Unlike previous results, our main new idea is the use of a discrete Grönwall lemma for the error estimate without any localization.

We study the large time behavior of entropic approximate solutions to one-dimensional, hyperbolic conservation laws with periodic initial data. Under mild assumptions on the numerical scheme, we prove the asymptotic convergence of the discrete solutions to a time- and space-periodic solution.

Solutions of initial-boundary value problems for systems of conservation laws depend on the underlying viscous mechanism, namely different viscosity operators lead to different limit solutions. Standard numerical schemes for approximating conservation laws do not take into account this fact and converge to solutions that are not necessarily physically relevant. We design numerical schemes that incorporate explicit information about the underlying viscosity mechanism and approximate the physical-viscosity solution. Numerical experiments illustrating the robust performance of these schemes are presented.

A set of fully nonlinear Boussinessq-type equations with improved linear and nonlinear properties is considered for wave–current interaction analysis. These phase-resolving equations are so that the highest order of the derivatives is three. We implement a new source function for the wave–current generation within the domain, which allows to generate a wide range of wave–current conditions. The set of equations is solved using a fourth order explicit numerical scheme which semi-discretizes the equations in space and then integrates in time using an explicit Runge–Kutta scheme. A novel treatment of the boundaries, which uses radiative boundary conditions for the current and damps the waves, is used to avoid boundary reflections. Several validation tests are presented to demonstrate the capabilities of the new model equations for wave–current interaction.

Tsunami run-up through the river is one of important features for better predictions and estimations of inundation characteristics. This study carried out numerical investigations of observed tsunami inundation around the Kido river especially focusing on the prediction of run-up speed of tsunami along the river. Based on non-linear long wave model, a sensitivity analysis was first carried out by changing several computational conditions such as tsunami profile, bottom frictions and the river discharge. It was found through this analysis that these conditions, within the range of expected uncertain variations, have certain influence on predicted run-up speed of tsunami. Second, this study also investigated the influence of the different discretization schemes of the model on predictive skills of the speed of tsunami run-up. Difference of conservative and non-conservative forms of non-linear term was investigated through numerical experiments of non-viscosity Burgers equations and it was found that the difference of these forms has significant influence on the predicted propagation speed of the bore. The same analysis was applied in the case of the Kido river and it was found that the tsunami run-up speed was increased up to 40% by selecting appropriate discretization schemes of conservative form.