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| {{Differential equations}}
| | Have we been thinking "how do I speed up my computer" lately? Well odds are should you are reading this article; then you are experiencing 1 of many computer issues which thousands of people discover that they face on a regular basis.<br><br>Windows Defender - this does come standard with countless Windows OS Machines, however otherwise will be download from Microsoft for free. It may enable safeguard against spyware.<br><br>So, this advanced dual scan is not merely one of the greater, however, it is very additionally freeware. And as of all of this which countless regard CCleaner among the better registry products in the market now. I would add that I personally prefer Regcure for the simple reason which it has a better interface and I know for a fact it is ad-ware without charge.<br><br>Handling intermittent errors - whenever there is a content to the effect which "memory or hard disk is malfunctioning", we may place inside new hardware to substitute the defective piece until the actual issue is found out. There are h/w diagnostic programs to identify the faulty portions.<br><br>Whenever it comes to software, this might be the vital part because it is the 1 running a system too as additional programs needed inside a works. Always maintain the cleanliness of the program from obsolete information by getting a superior [http://bestregistrycleanerfix.com/tune-up-utilities tuneup utilities]. Protect it from a virus found on the net by providing a workable virus protection system. You could moreover have a monthly clean up by running a defragmenter system. This technique it usually enhance the performance of your computer and for you to avoid any errors. If you think anything is incorrect with the software, plus we don't know how to fix it then refer to a technician.<br><br>Files with the DOC extension are also susceptible to viruses, but this will be solved by superior antivirus programs. Another issue is that .doc files could be corrupted, unreadable or damaged due to spyware, adware, and malware. These situations usually avoid users from correctly opening DOC files. This is when powerful registry cleaners become worthwhile.<br><br>In alternative words, if the PC has any corrupt settings inside the registry database, these settings might create a computer run slower and with a lot of mistakes. And regrettably, it's the case that XP is prone to saving many settings from the registry inside the incorrect means, making them unable to run correctly, slowing it down and causing a lot of errors. Each time you employ a PC, it has to read 100's of registry settings... plus there are usually a lot of files open at once which XP gets confuse plus saves numerous inside the incorrect means. Fixing these damaged settings may boost the speed of your program... plus to do that, we should look to utilize a 'registry cleaner'.<br><br>Fortunately, there's a easy method to fix virtually all a computer errors. You really should be able to fix corrupt registry files on the computer. And to do which, we will only use a tool known as a registry cleaner. These easy pieces of software actually scan by your PC and fix every corrupt file which could cause a issue to Windows. This enables the computer to employ all the files it wants, which not merely speeds it up - yet also stops all of the errors on a system too. |
| The '''finite-volume method''' (FVM) is a method for representing and evaluating [[partial differential equation]]s in the form of algebraic equations [LeVeque, 2002; Toro, 1999].
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| Similar to the [[finite difference method]] or [[finite element method]], values are calculated at discrete places on a meshed geometry. "Finite volume" refers to the small volume surrounding each node point on a mesh. In the finite volume method, volume integrals in a partial differential equation that contain a [[divergence]] term are converted to [[surface integral]]s, using the [[divergence theorem]]. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods are [[conservation law|conservative]]. Another advantage of the finite volume method is that it is easily formulated to allow for unstructured meshes. The method is used in many [[computational fluid dynamics]] packages.
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| ==1D example==
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| Consider a simple 1D [[advection]] problem defined by the following [[partial differential equation]]
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| :<math>\quad (1) \qquad \qquad \frac{\partial\rho}{\partial t}+\frac{\partial f}{\partial x}=0,\quad t\ge0.</math>
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| Here, <math> \rho=\rho \left( x,t \right) \ </math> represents the state variable and <math> f=f \left( \rho \left( x,t \right) \right) \ </math> represents the [[flux]] or flow of <math> \rho \ </math>. Conventionally, positive <math> f \ </math> represents flow to the right while negative <math> f \ </math> represents flow to the left. If we assume that equation (1) represents a flowing medium of constant area, we can sub-divide the spatial domain, <math> x \ </math>, into ''finite volumes'' or ''cells'' with cell centres indexed as <math> i \ </math>. For a particular cell, <math> i \ </math>, we can define the ''volume average'' value of <math> {\rho }_i \left( t \right) = \rho \left( x, t \right) \ </math> at time <math> {t = t_1 }\ </math> and <math>{ x \in \left[ x_{i-\frac{1}{2}} , x_{i+\frac{1}{2}} \right] }\ </math>, as
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| :<math>\quad (2) \qquad \qquad \bar{\rho}_i \left( t_1 \right) = \frac{1}{ x_{i+\frac{1}{2}} - x_{i-\frac{1}{2}}} \int_{x_{i-\frac{1}{2}}}^{x_{i+\frac{1}{2}}} \rho \left(x,t_1 \right)\, dx ,</math>
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| and at time <math> {t = t_2}\ </math> as,
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| :<math>\quad (3) \qquad \qquad \bar{\rho}_i \left( t_2 \right) = \frac{1}{x_{i+\frac{1}{2}} - x_{i-\frac{1}{2}}} \int_{x_{i-\frac{1}{2}}}^{x_{i+\frac{1}{2}}} \rho \left(x,t_2 \right)\, dx ,</math>
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| where <math> x_{i-\frac{1}{2}} \ </math> and <math> x_{i+\frac{1}{2}} \ </math> represent locations of the upstream and downstream faces or edges respectively of the <math> i^{th} \ </math> cell.
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| Integrating equation (1) in time, we have:
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| :<math>\quad (4) \qquad \qquad \rho \left( x, t_2 \right) = \rho \left( x, t_1 \right) - \int_{t_1}^{t_2} f_x \left( x,t \right)\, dt,</math>
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| where <math>f_x=\frac{\partial f}{\partial x}</math>.
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| To obtain the volume average of <math> \rho\left(x,t\right) </math> at time <math> t=t_{2} \ </math>, we integrate <math> \rho\left(x,t_2 \right) </math> over the cell volume, <math>\left[ x_{i-\frac{1}{2}} , x_{i+\frac{1}{2}} \right] </math> and divide the result by <math>\Delta x_i = x_{i+\frac{1}{2}}-x_{i-\frac{1}{2}} </math>, i.e.
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| :<math> \quad (5) \qquad \qquad \bar{\rho}_{i}\left( t_{2}\right) =\frac{1}{\Delta x_i}\int_{x_{i-\frac{1}{2}}}^{x_{i+\frac{1}{2}}}\left\{ \rho\left( x,t_{1}\right) - \int_{t_{1}}^{t_2} f_{x} \left( x,t \right) dt \right\} dx.</math>
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| We assume that <math> f \ </math> is well behaved and that we can reverse the order of integration. Also, recall that flow is normal to the unit area of the cell. Now, since in one dimension <math>f_x \triangleq \nabla f </math>, we can apply the [[divergence theorem]], i.e. <math>\oint_{v}\nabla\cdot fdv=\oint_{S}f\, dS </math>, and substitute for the volume integral of the [[divergence]] with the values of <math>f(x) \ </math> evaluated at the cell surface (edges <math>x_{i-\frac{1}{2}} \ </math> and <math> x_{i+\frac{1}{2}} \ </math>) of the finite volume as follows:
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| :<math>\quad (6) \qquad \qquad \bar{\rho}_i \left( t_2 \right) = \bar{\rho}_i \left( t_1 \right)
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| - \frac{1}{\Delta x_{i}}
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| \left( \int_{t_1}^{t_2} f_{i + \frac{1}{2}} dt
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| - \int_{t_1}^{t_2} f_{i - \frac{1}{2}} dt
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| \right) .</math>
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| where <math>f_{i \pm \frac{1}{2}} =f \left( x_{i \pm \frac{1}{2}}, t \right) </math>.
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| We can therefore derive a ''semi-discrete'' numerical scheme for the above problem with cell centres indexed as <math> i\ </math>, and with cell edge fluxes indexed as <math> i\pm\frac{1}{2} </math>, by differentiating (6) with respect to time to obtain:
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| :<math>\quad (7) \qquad \qquad \frac{d \bar{\rho}_i}{d t} + \frac{1}{\Delta x_i} \left[
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| f_{i + \frac{1}{2}} - f_{i - \frac{1}{2}} \right] =0 ,</math>
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| where values for the edge fluxes, <math> f_{i \pm \frac{1}{2}} </math>, can be reconstructed by interpolation or extrapolation of the cell averages. Equation (7) is ''exact'' for the volume averages; i.e., no approximations have been made during its derivation.
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| == General conservation law ==
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| We can also consider the general [[conservation law]] problem, represented by the following [[partial differential equation|PDE]],
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| :<math> \quad (8) \qquad \qquad {{\partial {\mathbf u}} \over {\partial t}} + \nabla \cdot {\mathbf f}\left( {\mathbf u } \right) = {\mathbf 0} . </math>
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| Here, <math> {\mathbf u} \ </math> represents a vector of states and <math>\mathbf f \ </math> represents the corresponding [[flux]] tensor. Again we can sub-divide the spatial domain into finite volumes or cells. For a particular cell, <math>i \ </math>, we take the volume integral over the total volume of the cell, <math>v _{i} \ </math>, which gives,
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| :<math> \quad (9) \qquad \qquad \int _{v_{i}} {{\partial {\mathbf u}} \over {\partial t}}\, dv
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| + \int _{v_{i}} \nabla \cdot {\mathbf f}\left( {\mathbf u } \right)\, dv = {\mathbf 0} .</math>
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| On integrating the first term to get the ''volume average'' and applying the ''divergence theorem'' to the second, this yields
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| :<math>\quad (10) \qquad \qquad
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| v_{i} {{d {\mathbf {\bar u} }_{i} } \over {dt}} + \oint _{S_{i} }
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| {\mathbf f} \left( {\mathbf u } \right) \cdot {\mathbf n }\ dS = {\mathbf 0}, </math>
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| where <math> S_{i} \ </math> represents the total surface area of the cell and <math>{\mathbf n}</math> is a unit vector normal to the surface and pointing outward. So, finally, we are able to present the general result equivalent to (8), i.e.
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| :<math> \quad (11) \qquad \qquad
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| {{d {\mathbf {\bar u} }_{i} } \over {dt}} + {{1} \over {v_{i}} } \oint _{S_{i} }
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| {\mathbf f} \left( {\mathbf u } \right)\cdot {\mathbf n }\ dS = {\mathbf 0} .</math>
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| Again, values for the edge fluxes can be reconstructed by interpolation or extrapolation of the cell averages. The actual numerical scheme will depend upon problem geometry and mesh construction. [[MUSCL scheme|MUSCL]] reconstruction is often used in [[high resolution scheme]]s where shocks or discontinuities are present in the solution.
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| Finite volume schemes are conservative as cell averages change through the edge fluxes. In other words, ''one cell's loss is another cell's gain''!
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| ==See also==
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| *[[Finite element method]]
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| *[[Flux limiter]]
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| *[[Godunov's scheme]]
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| *[[Godunov's theorem]]
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| *[[High-resolution scheme]]
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| *[[KIVA (Software)]]
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| *[[MIT General Circulation Model]]
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| *[[MUSCL scheme]]
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| *[[Sergei K. Godunov]]
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| *[[Total variation diminishing]]
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| *[[Finite volume method for unsteady flow]]
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| ==References==
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| *'''Eymard, R. Gallouët, T. R. Herbin, R.''' (2000) ''The finite volume method'' Handbook of Numerical Analysis, Vol. VII, 2000, p. 713–1020. Editors: P.G. Ciarlet and J.L. Lions.
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| *'''LeVeque, Randall''' (2002), ''Finite Volume Methods for Hyperbolic Problems'', Cambridge University Press.
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| *'''Toro, E. F.''' (1999), ''Riemann Solvers and Numerical Methods for Fluid Dynamics'', Springer-Verlag.
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| ==Further reading==
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| *'''Patankar, Suhas V.''' (1980), ''Numerical Heat Transfer and Fluid Flow'', Hemisphere.
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| *'''Hirsch, C.''' (1990), ''Numerical Computation of Internal and External Flows, Volume 2: Computational Methods for Inviscid and Viscous Flows'', Wiley.
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| *'''Laney, Culbert B.''' (1998), ''Computational Gas Dynamics'', Cambridge University Press.
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| *'''LeVeque, Randall''' (1990), ''Numerical Methods for Conservation Laws'', ETH Lectures in Mathematics Series, Birkhauser-Verlag.
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| *'''Tannehill, John C.''', et al., (1997), ''Computational Fluid mechanics and Heat Transfer'', 2nd Ed., Taylor and Francis.
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| *'''Wesseling, Pieter''' (2001), ''Principles of Computational Fluid Dynamics'', Springer-Verlag.
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| == External links ==
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| * [http://www.cmi.univ-mrs.fr/~herbin/PUBLI/bookevol.pdf The finite volume method] by R. Eymard, T Gallouët and R. Herbin, update of the article published in Handbook of Numerical Analysis, 2000
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| * [http://www.imtek.uni-freiburg.de/simulation/mathematica/IMSweb/imsTOC/Lectures%20and%20Tips/Simulation%20I/FVM_introDocu.html The Finite Volume Method (FVM) – An introduction] {{dead link|date=January 2010}} by Oliver Rübenkönig of [[Albert Ludwigs University of Freiburg]], available under the [[GNU Free Document License|GFDL]]. ({{wayback|url=http://www.imtek.uni-freiburg.de/simulation/mathematica/IMSweb/imsTOC/Lectures%20and%20Tips/Simulation%20I/FVM_introDocu.html}})
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| * [http://www.ctcms.nist.gov/fipy/ FiPy: A Finite Volume PDE Solver Using Python] from NIST.
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| * [http://depts.washington.edu/clawpack/ CLAWPACK]: a software package designed to compute numerical solutions to hyperbolic partial differential equations using a wave propagation approach
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| {{Numerical PDE}}
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| [[Category:Numerical differential equations]]
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| [[Category:Fluid dynamics]]
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| [[Category:Computational fluid dynamics]]
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| [[Category:Numerical analysis]]
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Have we been thinking "how do I speed up my computer" lately? Well odds are should you are reading this article; then you are experiencing 1 of many computer issues which thousands of people discover that they face on a regular basis.
Windows Defender - this does come standard with countless Windows OS Machines, however otherwise will be download from Microsoft for free. It may enable safeguard against spyware.
So, this advanced dual scan is not merely one of the greater, however, it is very additionally freeware. And as of all of this which countless regard CCleaner among the better registry products in the market now. I would add that I personally prefer Regcure for the simple reason which it has a better interface and I know for a fact it is ad-ware without charge.
Handling intermittent errors - whenever there is a content to the effect which "memory or hard disk is malfunctioning", we may place inside new hardware to substitute the defective piece until the actual issue is found out. There are h/w diagnostic programs to identify the faulty portions.
Whenever it comes to software, this might be the vital part because it is the 1 running a system too as additional programs needed inside a works. Always maintain the cleanliness of the program from obsolete information by getting a superior tuneup utilities. Protect it from a virus found on the net by providing a workable virus protection system. You could moreover have a monthly clean up by running a defragmenter system. This technique it usually enhance the performance of your computer and for you to avoid any errors. If you think anything is incorrect with the software, plus we don't know how to fix it then refer to a technician.
Files with the DOC extension are also susceptible to viruses, but this will be solved by superior antivirus programs. Another issue is that .doc files could be corrupted, unreadable or damaged due to spyware, adware, and malware. These situations usually avoid users from correctly opening DOC files. This is when powerful registry cleaners become worthwhile.
In alternative words, if the PC has any corrupt settings inside the registry database, these settings might create a computer run slower and with a lot of mistakes. And regrettably, it's the case that XP is prone to saving many settings from the registry inside the incorrect means, making them unable to run correctly, slowing it down and causing a lot of errors. Each time you employ a PC, it has to read 100's of registry settings... plus there are usually a lot of files open at once which XP gets confuse plus saves numerous inside the incorrect means. Fixing these damaged settings may boost the speed of your program... plus to do that, we should look to utilize a 'registry cleaner'.
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