The aim of this paper is to derive a posteriori error estimates for semilinear parabolic interface problems. More specifically, optimal order a posteriori error analysis in the - norm for semidiscrete semilinear parabolic interface problems is derived by using elliptic reconstruction technique introduced by Makridakis and Nochetto in (2003). A key idea for this technique is the use of error estimators derived for elliptic interface problems to obtain parabolic estimators that are of optimal order in space and time.

This paper aims to developed a high-order and accurate method for the solution of one-dimensional Lotka-Volterra-diffusion with Numman boundary conditions. A fourth-order compact finite difference scheme for spatial part combined with implicit-explicit Runge Kutta scheme in temporal are proposed. Furthermore, boundary points are discretized by using a compact finite difference scheme in terms of fourth order accuracy. A key idea for proposed scheme is to take full advantage of method of line (MOL), this is consequently enabling us to use implicit-explicit Runge Kutta method, that are of fourth order in time. We constructed fourth order accuracy in both space and time and is unconditionally stable. This is consequently leading to a reduction in the computational cost of the scheme. Numerical experiments show that the combination of the compact finite difference with IMEX- RK methods give an accurate and reliable for solving the Lotka-Volterra-diffusion.

The main goal of this paper is to developed a high-order and accurate method for the solution of one-dimensional of generalized Burgers-Fisher with Numman boundary conditions. We combined between a fourth-order compact finite difference scheme for spatial part with diagonal implicit Runge Kutta scheme in temporal part. In addition, we discretized boundary points by using a compact finite difference scheme in terms of fourth order accuracy. This combine leads to ordinary differential equation which will take full advantage of method of line (MOL). Some numerical experiments presented to show that the combination give an accurate and reliable for solving the generalized Burgers-Fisher problems.

This paper introduces a novel quartic B-spline collocation method to address the coupled Whitham–Broer–Kaup (WBK) problem. The WBK problem is a topic of interest in the study of nonlinear wave phenomena and has applications in various fields, including fluid dynamics, plasma physics, and nonlinear optics. The method combines spatial quartic B-spline scheme discretization, and Crank–Nicolson temporal discretization. It is unconditionally stable as proven by the Von-Neumann technique. Numerical examples demonstrate the method’s superior accuracy compared to existing solutions. Error analysis employs l 2 and l ∞ norms, while the method exhibits high computational efficiency. The nonlinearity is managed through Rubin-Graves linearization. Comparisons with prior approaches highlight its efficiency, stability, adaptability to complex problems. The quartic B-spline method is well-suited for simulating fluid flow phenomena in shallow water scenarios, offering high accuracy and low computational cost.