Discontinuous Galerkin approximation for excitatory-inhibitory networks with delay and refractory periods

2020 ◽  
Vol 31 (03) ◽  
pp. 2050041
Author(s):  
Dipty Sharma ◽  
Paramjeet Singh
Keyword(s):  
Discontinuous Galerkin ◽  
Galerkin Approximation ◽  
Strong Stability ◽  
Weno Scheme ◽  
Transmission Delays ◽  
Weighted Essentially Nonoscillatory ◽  
Refractory Periods ◽  
Dg Method ◽  
Analysis Of Stability ◽  
Inhibitory Networks

In this study, we consider the network of noisy leaky integrate-and-fire (NNLIF) model, which governs by a second-order nonlinear time-dependent partial differential equation (PDE). This equation uses the probability density approach to describe the behavior of neurons with refractory states and the transmission delays. A numerical approximation based on the discontinuous Galerkin (DG) method is used for the spatial discretization with the analysis of stability. The strong stability-preserving explicit Runge–Kutta (SSPERK) method is performed for the temporal discretization. Finally, some test examples and numerical simulations are given to examine the behavior of the solution. The execution of the constructed scheme is measured by the quantitative comparison with the existing finite difference technique, namely weighted essentially nonoscillatory (WENO) scheme.

Discontinuous Galerkin method for a nonlinear age-structured tumor cell population model with proliferating and quiescent phases

2020 ◽  
pp. 2150039
Author(s):  
Dipty Sharma ◽  
Paramjeet Singh
Keyword(s):  
Tumor Cell ◽  
Discontinuous Galerkin ◽  
Cell Population ◽  
Dynamical Behavior ◽  
Strong Stability ◽  
Weno Scheme ◽  
Tumor Cell Population ◽  
Dg Method ◽  
Age Structured ◽  
Age Structured Model

We propose a nonlinear age-structured model of tumor cell population with proliferating and quiescent phases. We apply the discontinuous Galerkin (DG) method to study its dynamical behavior. The DG numerical approximation is used for the spatial discretization and then the strong-stability-preserving explicit Runge–Kutta (SSPERK) method is performed for the temporal discretization. This paper aims to establish more efficient results in the sense of computational approach and compare these with analogous estimates for Weighted Essentially and Non-Oscillatory (WENO) scheme. Finally, some test examples and numerical simulations are given to illustrate theoretical results and to examine the behavior of the solution.

An equilibrated a posteriori error estimator for an Interior Penalty Discontinuous Galerkin approximation of the p-Laplace problem

2021 ◽  
Vol 36 (6) ◽  
pp. 313-336
Author(s):  
Ronald H. W. Hoppe ◽  
Youri Iliash
Keyword(s):  
Discontinuous Galerkin ◽  
Galerkin Approximation ◽  
Posteriori Error ◽  
Error Estimator ◽  
A Posteriori ◽  
A Posteriori Error Estimator ◽  
Interior Penalty ◽  
Posteriori Error Estimator ◽  
A Posteriori Error ◽  
Flux Function

Abstract We are concerned with an Interior Penalty Discontinuous Galerkin (IPDG) approximation of the p-Laplace equation and an equilibrated a posteriori error estimator. The IPDG method can be derived from a discretization of the associated minimization problem involving appropriately defined reconstruction operators. The equilibrated a posteriori error estimator provides an upper bound for the discretization error in the broken W 1,p norm and relies on the construction of an equilibrated flux in terms of a numerical flux function associated with the mixed formulation of the IPDG approximation. The relationship with a residual-type a posteriori error estimator is established as well. Numerical results illustrate the performance of both estimators.

A local discontinuous Galerkin approximation for systems with p-structure

2013 ◽  
Vol 34 (4) ◽  
pp. 1447-1488 ◽  
Author(s):  
L. Diening ◽  
D. Kroner ◽  
M. R  i ka ◽  
I. Toulopoulos
Keyword(s):  
Discontinuous Galerkin ◽  
Galerkin Approximation ◽  
Local Discontinuous Galerkin

scholarly journals Block Generalized Locally Toeplitz Sequences: From the Theory to the Applications

Axioms ◽  
2018 ◽  
Vol 7 (3) ◽  
pp. 49 ◽  
Author(s):  
Carlo Garoni ◽  
Mariarosa Mazza ◽  
Stefano Serra-Capizzano
Keyword(s):  
Finite Element ◽  
Differential Equations ◽  
Discontinuous Galerkin ◽  
Spectral Distribution ◽  
Galerkin Approximation ◽  
Higher Order ◽  
Numerical Discretization

The theory of generalized locally Toeplitz (GLT) sequences is a powerful apparatus for computing the asymptotic spectral distribution of matrices An arising from virtually any kind of numerical discretization of differential equations (DEs). Indeed, when the mesh fineness parameter n tends to infinity, these matrices An give rise to a sequence {An}n, which often turns out to be a GLT sequence or one of its “relatives”, i.e., a block GLT sequence or a reduced GLT sequence. In particular, block GLT sequences are encountered in the discretization of systems of DEs as well as in the higher-order finite element or discontinuous Galerkin approximation of scalar DEs. Despite the applicative interest, a solid theory of block GLT sequences has been developed only recently, in 2018. The purpose of the present paper is to illustrate the potential of this theory by presenting a few noteworthy examples of applications in the context of DE discretizations.

scholarly journals Consistent Weighted Average Flux of Well-Balanced TVD-RK Discontinuous Galerkin Method for Shallow Water Flows

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Thida Pongsanguansin ◽  
Montri Maleewong ◽  
Khamron Mekchay
Keyword(s):  
Shallow Water ◽  
Discontinuous Galerkin ◽  
Numerical Solutions ◽  
Order Approximation ◽  
Weighted Average ◽  
Flat Bottom ◽  
Flux Function ◽  
Dg Method ◽  
Average Flux ◽  
Steady Solutions

A well-balanced scheme with total variation diminishing Runge-Kutta discontinuous Galerkin (TVD-RK DG) method for solving shallow water equations is presented. Generally, the flux function at cell interface in the TVD-RK DG scheme is approximated by using the Harten-Lax-van Leer (HLL) method. Here, we apply the weighted average flux (WAF) which is higher order approximation instead of using the HLL in the TVD-RK DG method. The consistency property is shown. The modified well-balanced technique for flux gradient and source terms under the WAF approximations is developed. The accuracy of numerical solutions is demonstrated by simulating dam-break flows with the flat bottom. The steady solutions with shock can be captured correctly without spurious oscillations near the shock front. This presents the other flux approximations in the TVD-RK DG method for shallow water simulations.

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