Galerkin Time Research Articles

Mathematical modeling and computational analysis of hepatitis B virus transmission using the higher-order Galerkin scheme

Abstract Hepatitis B, a liver disease caused by the hepatitis B virus (HBV), poses a significant public health burden. The virus spreads through the exchange of bodily fluids between infected and susceptible individuals. Hepatitis B is a complex health challenge for individuals. In this research, we propose a nonlinear HBV mathematical model comprising seven compartments: susceptible, latent, acutely infected, chronically infected, carrier, recovered, and vaccinated individuals. Our model investigates the dynamics of HBV transmission and the impact of vaccination on disease control. Using the next-generation matrix approach, we derive the basic reproduction number R 0 {R}_{0} and determine the disease-free equilibrium points. We establish the global and local stability of the model using the Lyapunov function. The model is numerically solved using the higher-order Galerkin time discretization technique, and a comprehensive sensitivity analysis is carried out to investigate the impact of all physical parameters involved in the proposed nonlinear HBV mathematical model. A comparison was made of the accuracy and dependability with the findings produced using the Runge–Kutta fourth-order (RK4) approach. The findings highlight the critical need for vaccination, particularly among the exposed class, to facilitate rapid recovery and mitigate the spread of HBV. The results of this study provide valuable insights for public health policymakers and inform strategies for hepatitis B control and elimination.

Computational approaches on integrating vaccination and treatment strategies in the SIR model using Galerkin time discretization scheme

ABSTRACT In this manuscript, we carried out a thorough analysis of the general SIR model for epidemics. We broadened the model to include vaccination, treatment, and incidence rate. The vaccination rate is a testament to the alternatives made by individuals when it comes to receiving vaccinations and merging with the community of the recovered. The treatment rate measures how often people who have contracted a disease are able to transition into the recovered category. The continuous Galerkin-Petrov method, specifically the cGP(2)-method, is employed to calculate the numerical solutions of the models. The cGP(2)-method imperative to analyse two unknowns over every time interval. The unknowns can be determined by solving a block system of size (2 × 2). This approach demonstrates a strong level of precision throughout the entire time interval, with an impressive rate of convergence at the discrete time points. In addition, the article investigates the concept of the basic reproduction number ( R 0 ) and explores the intricacies of conducting sensitivity analysis of the developed system.

Boundary and distributed optimal control for a population dynamics PDE model with discontinuous in time Galerkin FEM schemes

We consider fully discrete finite element approximations for a semilinear optimal control system of partial differential equations in two cases: for distributed and Robin boundary control. The ecological predator-prey optimal control model is approximated by conforming finite element methods mimicking the spatial part, while a discontinuous Galerkin method is used for the time discretization. We investigate the sensitivity of the solution distance from the target function, in cases with smooth and rough initial data. We employ low, and higher-order polynomials in time and space whenever proper regularity is present. The approximation schemes considered are with and without control constraints, driving efficiently the system to desired states realized using non-linear gradient methods.

Efficient Improved Local Time Stepping with the Leapfrog Scheme for Transient 3-D Electromagnetic Analyses

An alternate boundary local time stepping (ABLTS) method is proposed for the discontinuous Galerkin time domain method for transient electromagnetic simulations to reduce the computation complexity of the local time stepping (LTS) method. The proposed method exhibites lower storage and time complexity than the conventional LF-LTS method . The stability, accuracy, and effectiveness of the ABLTS method are verified by applying it to the simulation of a resonator cavity and multi-layer microstrip antenna. The numerical results revealed that the developed method is effective for the transient electromagnetic simulation of antennas.

A Global-Local Error-Driven -Adaptive Scheme for Discontinuous Galerkin Time-Domain Method With Local Time-Stepping Strategy

In this paper, a highly efficient <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> -adaptive scheme has been developed for discontinuous Galerkin time domain (DGTD) method with local time stepping (LTS) strategy. In the proposed <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> -adaptive procedure, the dual-fold error driven scheme has been developed. Compared to the global error range corresponding to the preset base order range, the local maximal error in each discretized element is used to determine the allowable local range of the base order. The local error in each element is compared with the local error range to determine which operation, i.e., the base order increasing or decreasing, is implemented. The resulting two error thresholds are insensitive in the dynamic <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> -adaptive procedure. When applying the proposed <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> -adaptive scheme into the LTS method, a simple grouping approach in terms of the time steps related to the base orders has been developed to rapidly perform the marching-on-in time procedure. Several numerical examples including scattering from a missile, radiation of a ridge gap waveguide (RGW) antenna and transmission of a cylindrical cavity filter are given to demonstrate good accuracy and high efficiency of the proposed method.

Simulation of Electromagnetic Waves in Magnetized Cold Plasma by a Novel BZT-DGTD Method

This work presents a novel discontinuous Galerkin time domain (DGTD) approach based on second-order bilinear <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</i> transform (BZT) to analyze the behavior of electromagnetic waves in a magnetized cold plasma. The Maxwell’s equations and current density equation are utilized to formulate the governing mathematical expressions of magnetized plasma, and the BZT approach is employed to construct the time domain iterative formula. The suggested technique outperforms existing methods in terms of stability, performance, and aliasing effect avoidance. The efficacy of the suggested BZT-DGTD approach is investigated by analyzing electromagnetic wave propagation in unmagnetized and magnetized cold plasmas, and a comprehensive comparison is performed with commercial software WCT and COMSOL to validate its practicality. Furthermore, the features of electromagnetic wave propagation in plasma are investigated under numerous circumstances, and these instances are compatible with plasma physics.

The Effect of Nanoparticle Composition on the Surface-Enhanced Raman Scattering Performance of Plasmonic DNA Origami Nanoantennas.

A versatile generation of plasmonic nanoparticle dimers for surface-enhanced Raman scattering (SERS) is presented by combining a DNA origami nanofork and spherical and nonspherical Au or Ag nanoparticles. Combining different nanoparticle species with a DNA origami nanofork to form DNA origami nanoantennas (DONAs), the plasmonic nanoparticle dimers can be optimized for a specific excitation wavelength in SERS. The preparation of such nanoparticle dimers is robust enough to enable the characterization of SERS intensities and SERS enhancement factors of dye-modified DONAs on a single dimer level by measuring in total several thousands of dimers from five different dimer designs, each functionalized with three different Raman reporter molecules and measured at four different excitation wavelengths. Based on these data, SERS enhancement factor (EF) distributions have been determined for each dimer design and excitation wavelengths. The structures and measurement conditions with the highest EFs are suitable for single-molecule SERS (SM-SERS), which is realized by placing single dye molecules into hot spots. We demonstrate that the probability of placing single molecules in a strongly enhancing hot spot for SM-SERS can be increased by using anisotropic nanoparticles with several sharp edges, such as nanoflowers. Combining a Ag nanoparticle with a Au particle in one dimer structure allows for a broadband excitation covering almost the whole visible range. The most versatile plasmonic dimer structure for SERS combines a spherical Ag nanoparticle with a Au nanoflower. Employing the discontinuous Galerkin time domain method, we numerically investigate the bare, symmetric dimers with respect to spectral and near-field properties, showing that, indeed, the nanoflowers induce multiple hot spots located at the edges which surpass the intensity of the spherical dimers, indicating the possibility for SM-SERS. The presented DONA structures and SERS data provide a robust basis for applying such designs as versatile SERS tags and as substrates for SM-SERS measurements.

An Asymmetrical Mixed Higher-Order Discontinuous Galerkin Time Domain Method for Electromagnetic Scattering from the Plasma Sheath around a Hypersonic Vehicle

The plasma sheath during reentry of hypersonic vehicle is an unmagnetized and weakly ionized nonuniform plasma flow, which causes radio frequency blackout and strong plasma attenuation of electromagnetic wave. The physical properties of the nonuniform plasma flow were obtained using computational fluid dynamics software with unstructured grids. In this study, a detailed computational model was reconstructed with the high-order Lagrange grids for the nonuniform plasma flow region and the high-order Serendipity grids for the homogeneous medium region. In order to calculate the numerical flux between the two types of grids in the discontinuous Galerkin time domain (DGTD) algorithm, an asymmetric high-order element is constructed as a transition unit. Finally, the simulation results in the plasma sphere show that the above method improves the computational accuracy and decreases calculation. The amplitude and scattering about electromagnetic wave in nonuniform plasma flow are clarified in detail. It is suggested that the presented method could be an effective tool for investigating interaction between electromagnetic waves and plasma flow.

Temperature-drift effect analysis of microstrip filters based on parallel high-order DGTD and FETD method with memory reduction technique

The temperature-drift effect of microstrip filters is analyzed by the proposed electromagnetic-thermal co-simulation method based on parallel high-order discontinuous Galerkin time domain (DGTD) and finite-element time-domain (FETD) algorithms with a memory reduction technique. The element matrices of DGTD method are factorized into the product of coefficient matrices and universal matrices. The coefficient matrices are different for each tetrahedral element and need to be stored for all the elements. While the universal matrices are the same for all the tetrahedral elements and only need to be stored for one element. Since the sizes of the coefficient matrices are much smaller than the element matrices of DGTD method, the proposed method avoids storing the large element matrices and greatly reduce the memory requirement of the DGTD method. Thus it reduces the memory requirement of the whole electromagnetic-thermal co-simulation which is dominated by the memory consumption of the DGTD method. Large-scale parallel technique is adopted to accelerate the process of electromagnetic-thermal co-simulation. The proposed method provides a very powerful tool for temperature-drift effect analysis of microstrip filters.

Discontinuous Galerkin Time Stepping for Semilinear Parabolic Problems with Time Constant Delay

Discontinuous Galerkin Time Stepping for Semilinear Parabolic Problems with Time Constant Delay

A body-of-revolution discontinuous Galerkin time domain method based on curved mesh for Maxwell equations

ABSTRACT Numerical simulation of body-of-revolution (BoR) or axisymmetric structures for scattering and radiation problems has a broad spectrum of technology applications in microwave and optical engineering. In this work, we propose a high-order BoR discontinuous Galerkin time-domain method (BoR-DGTD) for solving Maxwell equations in the cylindrical coordinate system. The presented approach is based on or (m is the mode number and φ is the azimuth) expansions for different field components and thus yields a real-valued final system of equations. It saves half of the unknowns as compared to the existing method in which all the field components are assumed to have the same φ variation, i.e. . In contrast to the common practice of deriving variational formulation for the BoR Maxwell equations on two-dimensional grids in the meridian plane, we build it on three-dimensional coaxial toroidal subdomains. As a result, the singularity difficulty induced by the cylindrical coordinates is eliminated naturally. In addition, for BoR with curved material interfaces in the angular cross-section, a solution using curved grids in conjunction with the low-storage weight-adjusted approximation scheme is provided, which significantly enhances the simulation quality. Particularly, to achieve a set of optimized interpolation shape functions for the isoparametric transformation, an essential assistant in calculating the coefficient matrices, a heuristic procedure is developed to properly position the geometric control points on the curved grids. Some numerical experiments are conducted to demonstrate the accuracy and performance of our proposed BoR-DGTD method.

Advanced Numerical Methods for Graphene Simulation with Equivalent Boundary Conditions: A Review

Since the discovery of graphene, due to its excellent optical, thermal, mechanical and electrical properties, it has a broad application prospect in energy, materials, biomedicine, electromagnetism and other fields. A great quantity of researches on the physical mechanism of graphene has been applied to engineering in electromagnetism and optics. To study the properties of graphene, different kinds of numerical methods such as the mixed finite element method (Mixed FEM), the mixed spectral element method (Mixed SEM), Method of Auxiliary Sources (MAS), discontinuous Galerkin time-domain method (DGTD) and interior penalty discontinuous Galerkin time domain (IPDG) have been developed for simulating the electromagnetic field effects of graphene and equivalent boundary conditions such as impedance transmission boundary condition (ITBC), surface current boundary condition (SCBC), impedance matrix boundary condition (IMBC) and surface impedance boundary condition (SIBC) have been employed to replace graphene in the computational domain. In this work, the numerical methods with equivalent boundary conditions are reviewed, and some examples are provided to illustrate their applicability.

Multidimensional Generalized Riemann Problem Solver for Maxwell’s Equations

Approximate multidimensional Riemann solvers are essential building blocks in designing globally constraint-preserving finite volume time domain and discontinuous Galerkin time domain schemes for computational electrodynamics (CED). In those schemes, we can achieve high-order temporal accuracy with the help of Runge–Kutta or ADER time-stepping. This paper presents the design of a multidimensional approximate generalized Riemann problem (GRP) solver for the first time. The multidimensional Riemann solver accepts as its inputs the four states surrounding an edge on a structured mesh, and its output consists of a resolved state and its associated fluxes. In contrast, the multidimensional GRP solver accepts as its inputs the four states and their gradients in all directions; its output consists of the resolved state and its corresponding fluxes and the gradients of the resolved state. The gradients can then be used to extend the solution in time. As a result, we achieve second-order temporal accuracy in a single step. In this work, the formulation is optimized for linear hyperbolic systems with stiff, linear source terms because such a formulation will find maximal use in CED. Our formulation produces an overall constraint-preserving time-stepping strategy based on the GRP that is provably L-stable in the presence of stiff source terms. We present several stringent test problems, showing that the multidimensional GRP solver for CED meets its design accuracy and performs stably with optimal time steps. The test problems include cases with high conductivity, showing that the beneficial L-stability is indeed realized in practical applications.

Transmission dynamics of a novel HIV/AIDS model through a higher-order Galerkin time discretization scheme

There are numerous contagious diseases caused by pathogenic microorganisms, including bacteria, viruses, fungi, and parasites, that have the propensity to culminate in fatal consequences. A communicable disease is an illness caused by a contagion agent or its toxins and spread directly or indirectly to a susceptible animal or human host by an infected person, animal, vector, or immaterial environment. Human immunodeficiency virus (HIV) infection, hepatitis A, B, and C, and measles are all examples of communicable diseases. Acquired immunodeficiency syndrome (AIDS) is a communicable disease caused by HIV infection that has become the most severe issue facing humanity. The research work in this paper is to numerically explore a mathematical model and demonstrate the dynamics of HIV/AIDS disease transmission using a continuous Galerkin–Petrov time discretization of a higher-order scheme, specifically the cGP(2)-scheme. Depict a graphical and tabular comparison between the outcomes of the mentioned scheme and those obtained through other classical schemes that exist in the literature. Further, a comparison is performed relative to the well-known fourth-order Ruge–Kutta (RK4) method with different step sizes. By contrast, the suggested approach provided more accurate results with a larger step size than RK4 with a smaller step size. After validation and confirmation of the suggested scheme and code, we implement the method to the extended model by introducing a treatment rate and show the impact of various non-linear source terms for the generation of new cells. We also determined the basic reproduction number and use the Routh-Hurwitz criterion to assess the stability of disease-free and unique endemic equilibrium states of the HIV model.

Numerical study of light backscattering from layers of absorbing irregular particles larger than the wavelength

We simulate light backscattering from monolayers of absorbing irregular particles larger than the wavelength using a numerically exact discontinuous Galerkin time domain (DGTD) method. Varying the real and imaginary parts of the complex refractive index m, particle size and the structure of a monolayer we study the evolution of the negative polarization (NP) feature that is observed for many particulate surfaces near backscattering. Simulations show that the NP is enhanced when increasing absorption from Im(m)=0.06 to 0.3. The real part Re(m) plays little role in this case. We confirm correlation of NP with the particle size predicted by approximate models and observed in laboratory measurements. We also demonstrate that the interplay between single- and multiple scattering can be controlled by the topography of a monolayer resulting in enhanced or damped NP.

A discontinuous Galerkin time integration scheme for second order differential equations with applications to seismic wave propagation problems

A discontinuous Galerkin time integration scheme for second order differential equations with applications to seismic wave propagation problems

Superconvergent postprocessing of the discontinuous Galerkin time stepping method for nonlinear Volterra integro-differential equations

We propose a very simple but efficient postprocessing technique for improving the global accuracy of the discontinuous Galerkin (DG) time stepping method for solving nonlinear Volterra integro-differential equations. The key idea of the postprocessing technique is to add a higher order generalized Jacobi polynomial of degree k+1 with parameters (−1,0) to the DG approximation of degree k. We prove that the postprocessed DG approximations converge one order faster than the unprocessed DG approximations in the L2-, H1- and L∞-norms. Based on these postprocessed superconvergence results, we construct several a posteriori error estimators and prove that they are asymptotically exact as the step-size decreases. Numerical experiments are carried out to verify the theoretical results.

Space-time unfitted finite element methods for time-dependent problems on moving domains

We propose a space-time scheme that combines an unfitted finite element method in space with a discontinuous Galerkin time discretisation for the accurate numerical approximation of parabolic problems with moving domains or interfaces. We make use of an aggregated finite element space to attain robustness with respect to the cut locations. The aggregation is performed slab-wise to have a tensor product structure of the space-time discrete space, which is required in the numerical analysis. As an alternative, we also propose a space-time ghost penalty stabilisation term to attain robustness. We analyse the proposed algorithm, providing stability, condition number bounds and anisotropic a priori error estimates. A set of numerical experiments confirm the theoretical results for a parabolic problem on a moving domain. The method is applied for a mass transfer problem with changing topology.

Superconvergent Postprocessing of the Continuous Galerkin Time Stepping Method for Nonlinear Initial Value Problems with Application to Parabolic Problems

Superconvergent Postprocessing of the Continuous Galerkin Time Stepping Method for Nonlinear Initial Value Problems with Application to Parabolic Problems

A higher order Galerkin time discretization scheme for the novel mathematical model of COVID-19

&lt;abstract&gt; &lt;p&gt;In the present period, a new fast-spreading pandemic disease, officially recognised Coronavirus disease 2019 (COVID-19), has emerged as a serious international threat. We establish a novel mathematical model consists of a system of differential equations representing the population dynamics of susceptible, healthy, infected, quarantined, and recovered individuals. Applying the next generation technique, examine the boundedness, local and global behavior of equilibria, and the threshold quantity. Find the basic reproduction number $R_0$ and discuss the stability analysis of the model. The findings indicate that disease fee equilibria (DFE) are locally asymptotically stable when $R_0 &amp;lt; 1$ and unstable in case $R_0 &amp;gt; 1$. The partial rank correlation coefficient approach (PRCC) is used for sensitivity analysis of the basic reproduction number in order to determine the most important parameter for controlling the threshold values of the model. The linearization and Lyapunov function theories are utilized to identify the conditions for stability analysis. Moreover, solve the model numerically using the well known continuous Galerkin Petrov time discretization scheme. This method is of order 3 in the whole-time interval and shows super convergence of order 4 in the discrete time point. To examine the validity and reliability of the mentioned scheme, solve the model using the classical fourth-order Runge-Kutta technique. The comparison demonstrates the substantial consistency and agreement between the Galerkin-scheme and RK4-scheme outcomes throughout the time interval. Discuss the computational cost of the schemes in terms of time. The investigation emphasizes the precision and potency of the suggested schemes as compared to the other traditional schemes.&lt;/p&gt; &lt;/abstract&gt;