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.

scholarly journals Petrov-Galerkin Method for the Coupled Schrödinger-KdV Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
M. S. Ismail ◽  
Farida M. Mosally ◽  
Khadeejah M. Alamoudi
Keyword(s):  
Galerkin Method ◽  
Fourth Order ◽  
Kdv Equation ◽  
Approximation Technique ◽  
Implicit Midpoint Rule ◽  
Von Neumann ◽  
Midpoint Rule ◽  
B Splines ◽  
Runge Kutta Method ◽  
Second Order In Time

Petrov-Galerkin method is used to derive a numerical scheme for the coupled Schrödinger-KdV (SKdV) equations, where we have used the cubic B-splines as a test functions and a linear B-splines as a trial functions. Product approximation technique is used to deal with the nonlinear terms. An implicit midpoint rule and the Runge-Kutta method of fourth-order (RK4) are used to discretize in time. A block nonlinear pentadiagonal system is obtained. We solve this system by the fixed point method. The resulting scheme has a fourth-order accuracy in space direction and second-order in time direction in case of the implicit midpoint rule and it is unconditionally stable by von Neumann method. Using the RK4 method the scheme will be linear and fourth-order in time and space directions, and it is also conditionally stable. The exact soliton solution and the conserved quantities are used to assess the accuracy and to show the robustness and the efficiency of the proposed schemes.

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.

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.

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