scholarly journals An explicit unconditionally stable scheme: application to diffusive Covid-19 epidemic model

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yasir Nawaz ◽  
Muhammad Shoaib Arif ◽  
Kamaleldin Abodayeh ◽  
Wasfi Shatanawi
Keyword(s):  
Epidemic Model ◽  
General Type ◽  
Source Term ◽  
Time Dependent ◽  
Unconditionally Stable ◽  
Von Neumann ◽  
First Order ◽  
Von Neumann Stability ◽  
Von Neumann Stability Analysis ◽  
Stable Scheme

AbstractAn explicit unconditionally stable scheme is proposed for solving time-dependent partial differential equations. The application of the proposed scheme is given to solve the COVID-19 epidemic model. This scheme is first-order accurate in time and second-order accurate in space and provides the conditions to get a positive solution for the considered type of epidemic model. Furthermore, the scheme’s stability for the general type of parabolic equation with source term is proved by employing von Neumann stability analysis. Furthermore, the consistency of the scheme is verified for the category of susceptible individuals. In addition to this, the convergence of the proposed scheme is discussed for the considered mathematical model.

scholarly journals Runge-Kutta Integration of the Equal Width Wave Equation Using the Method of Lines

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
M. A. Banaja ◽  
H. O. Bakodah
Keyword(s):  
Wave Motion ◽  
Method Of Lines ◽  
Unconditionally Stable ◽  
Von Neumann ◽  
The Method Of Lines ◽  
Von Neumann Stability ◽  
Runge Kutta ◽  
Von Neumann Stability Analysis ◽  
Wave Phenomena ◽  
Good Agreement

The equal width (EW) equation governs nonlinear wave phenomena like waves in shallow water. Numerical solution of the (EW) equation is obtained by using the method of lines (MOL) based on Runge-Kutta integration. Using von Neumann stability analysis, the scheme is found to be unconditionally stable. Solitary wave motion and interaction of two solitary waves are studied using the proposed method. The three invariants of the motion are evaluated to determine the conservation properties of the generated scheme. Accuracy of the proposed method is discussed by computing theL2andL∞error norms. The results are found in good agreement with exact solution.

Stability of discrete schemes of Biot’s poroelastic equations

2020 ◽  
Author(s):  
Y Alkhimenkov ◽  
L Khakimova ◽  
Y Y Podladchikov
Keyword(s):  
Stability Analysis ◽  
Three Dimensions ◽  
Stability Conditions ◽  
Von Neumann ◽  
Explicit Schemes ◽  
Numerical Stability Analysis ◽  
Von Neumann Stability ◽  
Von Neumann Stability Analysis ◽  
Biot’S Equations ◽  
Implicit And Explicit

Summary The efficient and accurate numerical modeling of Biot’s equations of poroelasticity requires the knowledge of the exact stability conditions for a given set of input parameters. Up to now, a numerical stability analysis of the discretized elastodynamic Biot’s equations has been performed only for a few numerical schemes. We perform the von Neumann stability analysis of the discretized Biot’s equations. We use an explicit scheme for the wave propagation and different implicit and explicit schemes for Darcy’s flux. We derive the exact stability conditions for all the considered schemes. The obtained stability conditions for the discretized Biot’s equations were verified numerically in one-, two- and three-dimensions. Additionally, we present von Neumann stability analysis of the discretized linear damped wave equation considering different implicit and explicit schemes. We provide both the Matlab and symbolic Maple routines for the full reproducibility of the presented results. The routines can be used to obtain exact stability conditions for any given set of input material and numerical parameters.

Projected evolution superoperators and the density operator: theory and applications to inelastic scattering

1982 ◽  
Vol 60 (10) ◽  
pp. 1371-1386 ◽  
Author(s):  
R. E. Turner ◽  
J. S. Dahler ◽  
R. F. Snider
Keyword(s):  
Inelastic Scattering ◽  
Density Operator ◽  
Operator Method ◽  
Time Dependent ◽  
Interaction Picture ◽  
Von Neumann ◽  
First Order ◽  
Density Operators ◽  
Projection Operator Method ◽  
Time Dependent Solution

The projection operator method of Zwanzig and Feshbach is used to construct the time dependent density operator associated with a binary scattering event. The formula developed to describe this time dependence involves time-ordered cosine and sine projected evolution (memory) superoperators. Both Sehrödinger and interaction picture results are presented. The former is used to demonstrate the equivalence of the time dependent solution of the von Neumann equation and the more familiar, frequency dependent Laplaee transform solution. For two particular classes of projection superoperators projected density operators arc shown to be equivalent to projected wave functions. Except for these two special eases, no projected wave function analogs of projected density operators exist. Along with the decoupled-motions approximation, projected interaction picture density operators arc applied to inelastic scattering events. Simple illustrations arc provided of how this formalism is related to previously established results for two-state processes, namely, the theory of resonant transfer events, the first order Magnus approximation, and the Landau–Zener theory.

scholarly journals A Numerical Solution for Hirota-Satsuma Coupled KdV Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
M. S. Ismail ◽  
H. A. Ashi
Keyword(s):  
Vries Equation ◽  
Kdv Equation ◽  
Approximation Technique ◽  
Conserved Quantities ◽  
Test Functions ◽  
Order Accuracy ◽  
Implicit Midpoint Rule ◽  
Von Neumann ◽  
Von Neumann Stability ◽  
Von Neumann Stability Analysis

A Petrov-Galerkin method and product approximation technique are used to solve numerically the Hirota-Satsuma coupled Korteweg-de Vries equation, using cubicB-splines as test functions and a linearB-spline as trial functions. The implicit midpoint rule is used to advance the solution in time. Newton’s method is used to solve the block nonlinear pentadiagonal system we have obtained. The resulting schemes are of second order accuracy in both directions, space and time. The von Neumann stability analysis of the schemes shows that the two schemes are unconditionally stable. The single soliton solution and the conserved quantities are used to assess the accuracy and to show the robustness of the schemes. The interaction of two solitons, three solitons, and birth of solitons is also discussed.

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