Search

In this paper, we discuss a higher-order convergent numerical method for a two-parameter singularly perturbed differential equation with a discontinuous convection coefficient and a discontinuous source term. The presence of perturbation parameters generates boundary layers, and the discontinuous terms produce interior layers on both sides of the discontinuity. In order to obtain a higher-order convergent solution, a hybrid monotone finite difference scheme is constructed on a piecewise uniform Shishkin mesh, which is adapted inside the boundary and interior layers. On this mesh (including the point of discontinuity), the present method is almost second-order parameter-uniform convergent. The current scheme is compared with the standard upwind scheme, which is used at the point of discontinuity. The numerical experiments based on the proposed scheme show higher-order (almost second-order) accuracy compared to the standard upwind scheme, which provides almost first-order parameter-uniform convergence. [ABSTRACT FROM AUTHOR]

In this study, to solve the singularly perturbed delay convection–diffusion–reaction problem, we proposed a hybrid numerical scheme that converges uniformly. Parabolic right boundary layer outcomes from the presence of the small perturbation parameter. To grip this layer behaviour, the problem is solved by Bakhvalov–Shishkin mesh for spatial domain discretization and uniform mesh for temporal domain discretization. A hybrid scheme consisting of a non-polynomial spline scheme for fine mesh and a midpoint upwind scheme for coarse mesh is used to discretize the spatial derivative, while an implicit Euler scheme is used to discretize the time derivative. To make computed solutions more accurate and increase rate of convergence of the scheme, we applied Richardson extrapolation technique. The stability and convergence of the scheme are established. The scheme has a second order of convergence in the discrete supreme norm and is parametric uniformly convergent. The scheme’s application is demonstrated through two test problems. [ABSTRACT FROM AUTHOR]

In this paper, we discuss a higher-order convergent numerical method for a two-parameter singularly perturbed differential equation with a discontinuous convection coefficient and a discontinuous source term. The presence of perturbation parameters generates boundary layers, and the discontinuous terms produce interior layers on both sides of the discontinuity. In order to obtain a higher-order convergent solution, a hybrid monotone finite difference scheme is constructed on a piecewise uniform Shishkin mesh, which is adapted inside the boundary and interior layers. On this mesh (including the point of discontinuity), the present method is almost second-order parameter-uniform convergent. The current scheme is compared with the standard upwind scheme, which is used at the point of discontinuity. The numerical experiments based on the proposed scheme show higher-order (almost second-order) accuracy compared to the standard upwind scheme, which provides almost first-order parameter-uniform convergence.

ABSTRACTIn this study, to solve the singularly perturbed delay convection–diffusion–reaction problem, we proposed a hybrid numerical scheme that converges uniformly. Parabolic right boundary layer outcomes from the presence of the small perturbation parameter. To grip this layer behaviour, the problem is solved by Bakhvalov–Shishkin mesh for spatial domain discretization and uniform mesh for temporal domain discretization. A hybrid scheme consisting of a non-polynomial spline scheme for fine mesh and a midpoint upwind scheme for coarse mesh is used to discretize the spatial derivative, while an implicit Euler scheme is used to discretize the time derivative. To make computed solutions more accurate and increase rate of convergence of the scheme, we applied Richardson extrapolation technique. The stability and convergence of the scheme are established. The scheme has a second order of convergence in the discrete supreme norm and is parametric uniformly convergent. The scheme's application is demonstrated through two test problems.

An efficient numerical technique for a singularly perturbed parabolic reaction–diffusion problem with Robin type boundary conditions is presented in this work. The governing problem is discretized using the implicit Euler technique in time direction and a hybrid numerical technique that comprises a central finite difference method in the outer region and a cubic spline in compression method in the boundary layer regions in space direction. We use Shishkin-type meshes in the space domain to resolve the layers. The Robin type boundary conditions are handled using the second-order method. The totally discretized problem is well examined for stability and convergence. The use Bakhvalov–Shishkin and Vulanović–Shishkin meshes yields a more efficient and second-order convergence whereas Shishkin mesh produces almost second-order convergent solutions. Two test examples are computed. The current method has been compared to other methods in the scientific community.

This study aims at constructing a robust numerical scheme for solving singularly perturbed parabolic delay differential equations arising in the modeling of neuronal variability. Taylor's series expansion is applied to approximate the shift terms. The obtained result is approximated by using the implicit Euler method in the temporal discretization on a uniform step size with the hybrid numerical scheme consisting of the midpoint upwind method in the outer layer region and the cubic spline in tension method in the inner layer region on a piecewise uniform Shishkin mesh in the spatial discretization. The constructed scheme is shown to be an ε‐uniformly convergent accuracy of order OΛt+N−2ln3N. Two model examples are given to testify the theoretical findings. [ABSTRACT FROM AUTHOR]

We present a novel hybrid finite element‐spectral boundary integral (SBI) scheme that enables efficient simulation of earthquake cycles. This combined finite element‐SBI approach captures the benefits of finite elements in modeling problems with nonlinearities, as well as the computational superiority of SBI. The domain truncation enabled by this scheme allows us to utilize high‐resolution finite elements discretization to capture inhomogeneities or complexities that may exist in a narrow region surrounding the fault. Combined with an adaptive time stepping algorithm, this framework opens new opportunities for modeling earthquake cycles with high‐resolution fault zone physics. In this initial study, we consider a two‐dimensional antiplane model with a vertical strike‐slip fault governed by rate and state friction in the quasi‐dynamic limit under the radiation damping approximation. The proposed approach is first verified using the benchmark problem BP‐1 from the Southern California Earthquake Center sequence of earthquake and aseismic slip community verification effort. The computational framework is then utilized to model the earthquake sequence and aseismic slip of a fault embedded within a low‐velocity fault zone (LVFZ) with different widths and compliance levels. Our results indicate that sufficiently compliant LVFZs contribute to the emergence of subsurface events that fail to penetrate to the free surface and may experience earthquake clusters with nonuniform interseismic time. Furthermore, the LVFZ leads to slip rate amplification relative to the homogeneous elastic case. We discuss the implications of our results for understanding earthquake complexity as an interplay of fault friction and bulk heterogeneities. Key Points: We develop a hybrid finite element and spectral boundary integral scheme to efficiently simulate earthquake cyclesLow‐velocity fault zone changes the behavior of earthquake sequence and aseismic slipSubsurface events emerge for sufficiently compliant low‐velocity fault zones and lead to slip deficit [ABSTRACT FROM AUTHOR]

When metallic (or plasmonic) nanospheres are nearly touching, strong concentration of light can occur in the narrow gap regions. This phenomenon has potential applications in nanophotonics, biosensing, and spectroscopy. Understanding the strong interaction between the plasmonic spheres turns out to be quite challenging; indeed, an extremely high computational cost is required to compute the electromagnetic field. Also, the classical method of image charges, which is effective for the dielectric spheres system, is not valid for plasmonic spheres because of their negative permittivities. Here we develop new analytical and numerical methods for the plasmonic spheres system by clarifying the connection between transformation optics and the method of image charges. We derive fully analytic solutions valid for two plasmonic spheres. We then develop a hybrid numerical scheme for computing the field distribution produced by an arbitrary number of spheres. Our method is highly efficient and accurate even in the nearly touching case and is valid for plasmonic spheres. [ABSTRACT FROM AUTHOR]

Abstract: A nonlinear diffusion problem for drug release from 2D polymeric systems with finite dissolution rate is considered. The numerical solutions of such problems were obtained under some constraints in respect to the system geometry, the type of adsorption isotherm and concentration dependent diffusivity [Frenning, G., Stromme, M., 2003. Drug release modeled by dissolution, diffusion and immobilization. Int. J. Pharm. 250, 137–145; Frenning, G., Brohede, U., Stromme, M., 2005. Finite element analysis of the release of slowly dissolving drugs from cylindrical matrix systems. J. Control. Release 107, 320–329]. It is derived a numerical approach to solving a generalized problem, which avoids the above limitations. The proposed numerical scheme based on finite element domain approximation and time difference method is used for simulation of 2D drug release under various model parameters. The effects of drug adsorption and concentration dependent drug diffusivity are demonstrated. [Copyright &y& Elsevier]