Application of a Hybrid Method to the Solution of the Nonlinear Burgers’ Equation

2003 ◽  
Vol 70 (6) ◽  
pp. 926-929 ◽  
Author(s):  
Bor-Lih Kuo ◽  
Chao-Kuang Chen
Keyword(s):  
Reynolds Number ◽  
Numerical Methods ◽  
Finite Difference ◽  
Hybrid Method ◽  
Burgers Equation ◽  
Initial Conditions ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Approximation Techniques ◽  
Good Agreement

This paper presents the use of a hybrid method which combines differential transformation and finite difference approximation techniques in the solution of the nonlinear Burgers’ equation for various values of Reynolds number including high values. In order to demonstrate the accuracy and validity of the proposed method, it is used to solve several examples of Burgers’ equation, with each example having different initial conditions and boundary conditions. It is found that the results obtained are in good agreement with the analytical solutions, and that the results are more accurate than those provided by other approximate numerical methods.

Differential conduction at axonal bifurcations. II. Theoretical basis

1988 ◽  
Vol 59 (4) ◽  
pp. 1286-1295 ◽  
Author(s):  
N. Stockbridge
Keyword(s):  
Numerical Methods ◽  
Finite Difference ◽  
Theoretical Basis ◽  
Axial Current ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Cable Equation ◽  
Frequency Dependent ◽  
Membrane Area ◽  
Conical Section

1. Numerical methods are employed to describe unbranched and branched axons like those employed in the previous paper (13). The model axon consists of a set of conical-section membrane patches having identical Hodgkin-Huxley (6) properties and which are connected by a finite-difference approximation to the cable equation (7). 2. Frequency-dependent differential conduction is shown to occur in both unbranched and branched axons in the model, much as observed experimentally. 3. The effect is shown to occur when one limb is electrically shorter than another and results from differences in the way in which axial current entering such branches is distributed between membrane area near the bifurcation and membrane area far away.

Vortex breakdown mechanics of a laminar confined swirling flow with large expansion ratio using a non-uniform finite difference approximation

2021 ◽  
pp. 095440622110071
Author(s):  
F-J Granados-Ortiz ◽  
L Rodríguez-Tembleque ◽  
J Ortega-Casanova
Keyword(s):  
Reynolds Number ◽  
Finite Difference ◽  
Vortex Breakdown ◽  
Swirling Flow ◽  
Critical Value ◽  
Expansion Ratio ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Large Expansion ◽  
Swirl Parameter

Abrupt expansions are a very frequent geometry in mechanical engineering systems, i.e. in combustion chambers, valves, heat exchangers or impinging cooling devices. However, despite the large number of devices that use this geometry, the expanded flow behaviour still needs further research to understand and predict the full system performance. This paper presents the application of the non-uniform finite difference approximation method developed in Sanmiguel et al. for the numerical characterisation of a confined swirling laminar jet discharging with a large expansion ratio. This investigation can be considered an extension of previous work by Revuelta, but now a swirling flow is generated by a rotating pipe upstream the expansion. The structures found when a fully-developed rotating Hagen-Poiseuille flow discharges into a much larger pipe section are summarised in a bifurcation diagram, whose coordinates are the Reynolds number of the jet ( Rej) and the swirl parameter ( L), for which the time-dependent, axisymmetric and incompressible Navier-Stokes equations are integrated numerically. For values of the jet Reynolds number below 200, there is a critical value of the swirl parameter above which stable vortex breakdown appears. For values of the Reynolds number above 200, three different behaviours are observed, and each performance appears for a critical value of the swirl parameter. When increasing the swirl parameter from zero, the flow becomes axisymmetrically unstable, showing an oscillatory behaviour. If further increasing the swirl intensity, the oscillatory flow coexists with a vortex breakdown bubble and, finally, a steady vortex breakdown is reached. The expansion ratio ε considered in all the simulations is 1[Formula: see text]. In previous literature, the exactness of the limiting critical Rej and L values that define these behaviours has been found to be influenced by the variability in the inlet profile conditions, which affects the expanded flow. This enhances the importance in the present investigation to accurately simulate the discharge pipe inlet profiles.

scholarly journals Stable and Efficient Modeling of Anelastic Attenuation in Seismic Wave Propagation

2012 ◽  
Vol 12 (1) ◽  
pp. 193-225 ◽  
Author(s):  
N. Anders Petersson ◽  
Björn Sjögreen
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Half Space ◽  
Material Model ◽  
Second Order ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Space Problem ◽  
Standard Linear Solid ◽  
Standard Linear

AbstractWe develop a stable finite difference approximation of the three-dimensional viscoelastic wave equation. The material model is a super-imposition of N standard linear solid mechanisms, which commonly is used in seismology to model a material with constant quality factor Q. The proposed scheme discretizes the governing equations in second order displacement formulation using 3N memory variables, making it significantly more memory efficient than the commonly used first order velocity-stress formulation. The new scheme is a generalization of our energy conserving finite difference scheme for the elastic wave equation in second order formulation [SIAM J. Numer. Anal., 45 (2007), pp. 1902-1936]. Our main result is a proof that the proposed discretization is energy stable, even in the case of variable material properties. The proof relies on the summation-by-parts property of the discretization. The new scheme is implemented with grid refinement with hanging nodes on the interface. Numerical experiments verify the accuracy and stability of the new scheme. Semi-analytical solutions for a half-space problem and the LOH.3 layer over half-space problem are used to demonstrate how the number of viscoelastic mechanisms and the grid resolution influence the accuracy. We find that three standard linear solid mechanisms usually are sufficient to make the modeling error smaller than the discretization error.

Sign in / Sign up

Export Citation Format

Share Document