scholarly journals Recent Developments on Operator-Difference Schemes for Solving Nonlocal BVPs for the Wave Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-14 ◽  
Author(s):  
Mehmet Emir Koksal
Keyword(s):  
Difference Scheme ◽  
Nonlocal Boundary ◽  
Second Order ◽  
Difference Schemes ◽  
Hyperbolic Partial Differential Equations ◽  
Operator Coefficient ◽  
Processing Unit ◽  
Order Difference ◽  
First Order ◽  
Central Processing

The second-order one-dimensional linear hyperbolic equation with time and space variable coefficients and nonlocal boundary conditions is solved by using stable operator-difference schemes. Two new second-order difference schemes recently appeared in the literature are compared numerically with each other and with the rather old first-order difference scheme all to solve abstract Cauchy problem for hyperbolic partial differential equations with time-dependent unbounded operator coefficient. These schemes are shown to be absolutely stable, and the numerical results are presented to compare the accuracy and the execution times. It is naturally seen that the second-order difference schemes are much more advantages than the first-order ones. Although one of the second-order difference scheme is less preferable than the other one according to CPU (central processing unit) time consideration, it has superiority when the accuracy weighs more importance.

Second-Order Accurate Explicit Finite-Difference Schemes for Waterhammer Analysis

1985 ◽  
Vol 107 (4) ◽  
pp. 523-529 ◽  
Author(s):  
M. H. Chaudhry ◽  
M. Y. Hussaini
Keyword(s):  
Finite Difference ◽  
Computer Time ◽  
Second Order ◽  
Difference Schemes ◽  
Hyperbolic Partial Differential Equations ◽  
Finite Difference Schemes ◽  
Piping System ◽  
Characteristic Method ◽  
First Order ◽  
Explicit Finite Difference

Three second-order accurate explicit finite-difference schemes—MacCormack’s method, Lambda scheme and Gabutti scheme—are introduced to solve the quasilinear, hyperbolic partial differential equations describing waterhammer phenomenon in closed conduits. The details of these schemes and the treatment of boundary conditions are presented. The results computed by using these schemes for a simple frictionless piping system are compared with the exact solution. It is shown that for the same accuracy, second-order schemes require fewer computational nodes and less computer time as compared to those required by the first-order characteristic method.

scholarly journals Finite Difference Method for Hyperbolic Equations with the Nonlocal Integral Condition

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Necmettin Aggez
Keyword(s):  
Space Variable ◽  
Hyperbolic Equations ◽  
Nonlocal Boundary ◽  
Integral Condition ◽  
Variable Coefficients ◽  
Difference Schemes ◽  
Order Difference ◽  
One Dimensional ◽  
Nonlocal Integral Condition ◽  
Multidimensional Hyperbolic Equations

The stable difference schemes for the approximate solution of the nonlocal boundary value problem for multidimensional hyperbolic equations with dependent in space variable coefficients are presented. Stability of these difference schemes and of the first- and second-order difference derivatives is obtained. The theoretical statements for the solution of these difference schemes for one-dimensional hyperbolic equations are supported by numerical examples.

A Numerical Simulation Algorithm of the Inviscid Dynamics of Axisymmetric Swirling Flows in a Pipe

2016 ◽  
Vol 138 (9) ◽  
Author(s):  
J. Granata ◽  
L. Xu ◽  
Z. Rusak ◽  
S. Wang
Keyword(s):  
Numerical Simulation ◽  
Reynolds Number ◽  
Difference Scheme ◽  
Second Order ◽  
Swirling Flows ◽  
Inviscid Limit ◽  
Simulation Algorithm ◽  
Spatial Integration ◽  
Order Difference ◽  
Breakdown Process

Current simulations of swirling flows in pipes are limited to relatively low Reynolds number flows (Re < 6000); however, the characteristic Reynolds number is much higher (Re > 20,000) in most of engineering applications. To address this difficulty, this paper presents a numerical simulation algorithm of the dynamics of incompressible, inviscid-limit, axisymmetric swirling flows in a pipe, including the vortex breakdown process. It is based on an explicit, first-order difference scheme in time and an upwind, second-order difference scheme in space for the time integration of the circulation and azimuthal vorticity. A second-order Poisson equation solver for the spatial integration of the stream function in terms of azimuthal vorticity is used. In addition, when reversed flow zones appear, an averaging step of properties is applied at designated time steps. This adds slight artificial viscosity to the algorithm and prevents growth of localized high-frequency numerical noise inside the breakdown zone that is related to the expected singularity that must appear in any flow simulation based on the Euler equations. Mesh refinement studies show agreement of computations for various mesh sizes. Computed examples of flow dynamics demonstrate agreement with linear and nonlinear stability theories of vortex flows in a finite-length pipe. Agreement is also found with theoretically predicted steady axisymmetric breakdown states in a pipe as flow evolves to a time-asymptotic state. These findings indicate that the present algorithm provides an accurate prediction of the inviscid-limit, axisymmetric breakdown process. Also, the numerical results support the theoretical predictions and shed light on vortex dynamics at high Re.

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