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Central differencing scheme
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<h2 id="steady-state-convection-diffusion-equation">Steady-state convection diffusion equation</h2> <p>The <a href="/facts/Convection%25E2%2580%2593diffusion_equation/jHoHG5Xj">convection–diffusion equation</a> is a collective representation of diffusion and convection equations, and describes or explains every physical phenomenon involving convection and diffusion in the transference of particles, energy and other physical quantities inside a physical system:<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p><p> div ( ρ u φ ) = div ( Γ ∇ φ ) + S φ ; {\displaystyle \operatorname {div} (\rho u\varphi )=\operatorname {div} (\Gamma \nabla \varphi )+S_{\varphi };\,} where Г is <a href="/facts/Diffusion_coefficient/GSngNem0">diffusion coefficient</a> and Φ is the <a href="/facts/Property/F6bGqY2w">property</a>. </p> <h2 id="formulation-of-steady-state-convection-diffusion-equation">Formulation of steady-state convection diffusion equation</h2> <p>Formal <a href="/facts/Integral/V7lyV997">integration</a> of steady-state convection–diffusion equation over a <a href="/facts/Control_volume/vQT19btg">control volume</a> gives </p> <table><tbody><tr><td> ∫ A n ⋅ ( ρ u φ ) d A = ∫ A n ⋅ ( Γ ∇ φ ) d A + ∫ C V S φ d V {\displaystyle \int _{A}\,n\cdot (\rho u\varphi )\,dA=\int _{A}\,n\cdot (\Gamma \nabla \varphi )\,dA+\int _{CV}\,S_{\varphi }\,dV} </td> <td></td> <td>Equation 1</td></tr></tbody></table> <p>This equation represents flux balance in a control volume. The left side gives the net convective flux, and the right side contains the net diffusive flux and the generation or destruction of the property within the control volume. </p><p>In the absence of source term equation, one becomes </p> <table><tbody><tr><td> d d x ( ρ u φ ) = d d x ( d φ d x ) {\displaystyle {\frac {d}{dx}}(\rho u\varphi )={\frac {d}{dx}}\left({\frac {d\varphi }{dx}}\right)} </td> <td></td> <td>Equation 2</td></tr></tbody></table> <p><a href="/facts/Continuity_equation/GVav2Vet">Continuity equation</a>: </p> <table><tbody><tr><td> d d x ( ρ u ) = 0 {\displaystyle {d \over dx}(\rho u)=0} </td> <td></td> <td>Equation 3</td></tr></tbody></table> <p>Assuming a control volume and integrating equation 2 over control volume gives: </p> <table><tbody><tr><td> ( ρ u φ A ) e − ( ρ u φ A ) w = ( Γ A d φ / d x ) e − ( Γ A d φ / d x ) w {\displaystyle (\rho u\varphi A)_{e}-(\rho u\varphi A)_{w}=(\Gamma Ad\varphi /dx)_{e}-(\Gamma Ad\varphi /dx)_{w}} </td> <td></td> <td>Integrated convection–diffusion equation</td></tr></tbody></table> <p>Integration of equation 3 yields: </p> <table><tbody><tr><td> ( ρ u A ) e − ( ρ u A ) w = 0 {\displaystyle (\rho uA)_{e}-(\rho uA)_{w}=0} </td> <td></td> <td>Integrated continuity equation</td></tr></tbody></table> <p>It is convenient to define two variables to represent the convective mass flux per unit area and diffusion conductance at cell faces, for example: F = ρ u {\displaystyle F=\rho u} D = Γ / δ x {\displaystyle D=\Gamma /\delta x} </p><p>Assuming A e = A w {\displaystyle A_{e}=A_{w}} , we can write integrated convection–diffusion equation as: F e φ e − F w φ w = D e ( φ E − φ P ) − D w ( φ P − φ W ) {\displaystyle F_{e}\varphi _{e}-F_{w}\varphi _{w}=D_{e}(\varphi _{E}-\varphi _{P})-D_{w}(\varphi _{P}-\varphi _{W})} </p><p>And integrated continuity equation as: F e − F w = 0 {\displaystyle F_{e}-F_{w}=0} </p><p>In a central differencing scheme, we try linear interpolation to compute cell face values for convection terms. </p><p>For a uniform grid, we can write cell face values of property Φ as φ e = 1 2 ( φ E + φ P ) , φ w = 1 2 ( φ P + φ W ) {\displaystyle \varphi _{e}={\tfrac {1}{2}}(\varphi _{E}+\varphi _{P}),\quad \varphi _{w}={\tfrac {1}{2}}(\varphi _{P}+\varphi _{W})} </p><p>On substituting this into integrated convection-diffusion equation, we obtain: F e φ E + φ P 2 − F w φ W + φ P 2 = D e ( φ E − φ P ) − D w ( φ P − φ W ) {\displaystyle F_{e}{\frac {\varphi _{E}+\varphi _{P}}{2}}-F_{w}{\frac {\varphi _{W}+\varphi _{P}}{2}}=D_{e}(\varphi _{E}-\varphi _{P})-D_{w}(\varphi _{P}-\varphi _{W})} </p><p>And on rearranging: [ ( D w + F w 2 ) + ( D e − F e 2 ) + ( F e − F w ) ] φ P = ( D w + F w 2 ) φ W + ( D e − F e 2 ) φ E {\displaystyle \left[\left(D_{w}+{\frac {F_{w}}{2}}\right)+\left(D_{e}-{\frac {F_{e}}{2}}\right)+(F_{e}-F_{w})\right]\varphi _{P}=\left(D_{w}+{\frac {F_{w}}{2}}\right)\varphi _{W}+\left(D_{e}-{\frac {F_{e}}{2}}\right)\varphi _{E}} a P φ P = a W φ W + a E φ E {\displaystyle a_{P}\varphi _{P}=a_{W}\varphi _{W}+a_{E}\varphi _{E}} </p> <h2 id="different-aspects-of-central-differencing-scheme">Different aspects of central differencing scheme</h2> <h3>Conservativeness</h3> <p>Conservation is ensured in central differencing scheme since overall flux balance is obtained by summing the net flux through each control volume taking into account the boundary fluxes for the control volumes around nodes 1 and 4. </p> <p>Boundary flux for control volume around node 1 and 4 [ Γ e 1 ( φ 2 − φ 1 ) δ x − q A ] + [ Γ e 2 ( φ 3 − φ 2 ) δ x − Γ w 2 ( φ 2 − φ 1 ) δ x ] + [ Γ e 3 ( φ 4 − φ 3 ) δ x − Γ w 3 ( φ 3 − φ 2 ) δ x ] + [ q B − Γ w 4 ( φ 4 − φ 3 ) δ x ] = q B − q A {\displaystyle {\begin{aligned}&\left[{\frac {\Gamma _{e_{1}}(\varphi _{2}-\varphi _{1})}{\delta x}}-q_{A}\right]+\left[{\frac {\Gamma _{e_{2}}(\varphi _{3}-\varphi _{2})}{\delta x}}-{\frac {\Gamma _{w_{2}}(\varphi _{2}-\varphi _{1})}{\delta x}}\right]\\[10pt]+{}&\left[{\frac {\Gamma _{e_{3}}(\varphi _{4}-\varphi _{3})}{\delta x}}-{\frac {\Gamma _{w_{3}}(\varphi _{3}-\varphi _{2})}{\delta x}}\right]+\left[q_{B}-{\frac {\Gamma _{w_{4}}(\varphi _{4}-\varphi _{3})}{\delta x}}\right]=q_{B}-q_{A}\end{aligned}}} because Γ e 1 = Γ w 2 , Γ e 2 = Γ w 3 , Γ e 3 = Γ w 4 {\displaystyle \Gamma _{e_{1}}=\Gamma _{w_{2}},\Gamma _{e_{2}}=\Gamma _{w_{3}},\Gamma _{e_{3}}=\Gamma _{w_{4}}} </p> <h3>Boundedness</h3> <p>Central differencing scheme satisfies first condition of <a href="/facts/Bounded_set/yCldjtoy">boundedness</a>. </p><p>Since F e − F w = 0 {\displaystyle F_{e}-F_{w}=0} from continuity equation, therefore; a P φ P = a W φ W + a E φ E {\displaystyle a_{P}\varphi _{P}=a_{W}\varphi _{W}+a_{E}\varphi _{E}} </p><p>Another essential requirement for boundedness is that all coefficients of the discretised equations should have the same sign (usually all positive). But this is only satisfied when (<a href="/facts/Peclet_number/lIGehNcg">peclet number</a>) F e / D e < 2 {\displaystyle F_{e}/D_{e}<2} because for a unidirectional flow ( F e > 0 , F w > 0 {\displaystyle F_{e}>0,F_{w}>0} ) a E = ( D e − F e / 2 ) {\displaystyle a_{E}=(D_{e}-F_{e}/2)} is always positive if D e > F e / 2 {\displaystyle D_{e}>F_{e}/2} </p> <h3>Transportiveness</h3> <p>It requires that transportiveness changes according to magnitude of peclet number i.e. when pe is zero φ {\displaystyle \varphi } is spread in all directions equally and as Pe increases (convection > diffusion) φ {\displaystyle \varphi } at a point largely depends on upstream value and less on downstream value. But central differencing scheme does not possess transportiveness at higher pe since Φ at a point is average of neighbouring nodes for all Pe. </p> <h3>Accuracy</h3> <p>The <a href="/facts/Taylor_series/M0aTuftH">Taylor series</a> truncation error of the central differencing scheme is second order. Central differencing scheme will be accurate only if Pe < 2. Owing to this limitation, central differencing is not a suitable discretisation practice for general purpose flow calculations. </p> <h2 id="applications-of-central-differencing-schemes">Applications of central differencing schemes</h2> <ul><li>They are currently used on a regular basis in the solution of the <a href="/facts/Euler_equations/F1zh068s">Euler equations</a> and <a href="/facts/Navier%25E2%2580%2593Stokes_equations/48pQugEa">Navier–Stokes equations</a>.</li> <li>Results using central differencing approximation have shown noticeable improvements in accuracy in smooth regions.</li> <li><a href="/facts/Shock_wave/CqrG1Xm5">Shock wave</a> representation and <a href="/facts/Boundary-layer/go8gqWM1">boundary-layer</a> definition can be improved on coarse meshes.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a></li></ul> <h2 id="advantages">Advantages</h2> <ul><li>Simpler to program, requires less computer time per step, and works well with multigrid <a href="/facts/Acceleration/UVDmjYvR">acceleration</a> techniques</li> <li>Has a free parameter in conjunction with the fourth-difference dissipation, which is needed to approach a steady state.</li> <li>More accurate than the first-order upwind scheme if the Peclet number is less than 2.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a></li></ul> <h2 id="disadvantages">Disadvantages</h2> <ul><li>Somewhat more dissipative</li> <li>Leads to <a href="/facts/Oscillations/6vSrm84a">oscillations</a> in the solution or divergence if the local Peclet number is larger than 2.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a></li></ul> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Finite_difference_method/EoG81Qn9">Finite difference method</a></li> <li><a href="/facts/Finite_difference/aY1txFhj">Finite difference</a></li> <li><a href="/facts/Taylor_series/M0aTuftH">Taylor series</a></li> <li><a href="/facts/Taylor_theorem/gxQTaNQF">Taylor theorem</a></li> <li><a href="/facts/Convection%25E2%2580%2593diffusion_equation/jHoHG5Xj">Convection–diffusion equation</a></li> <li><a href="/facts/Diffusion/KbAxroBa">Diffusion</a></li> <li><a href="/facts/Convection/E8vsCPhu">Convection</a></li> <li><a href="/facts/Peclet_number/lIGehNcg">Peclet number</a></li> <li><a href="/facts/Linear_interpolation/galkvRGy">Linear interpolation</a></li> <li><a href="/facts/Symmetric_derivative/aF83At0g">Symmetric derivative</a></li> <li><a href="/facts/Upwind_differencing_scheme_for_convection/j2jo7gIB">Upwind differencing scheme for convection</a></li></ul> <h2 id="further-reading">Further reading</h2> <ul><li><i>Computational Fluid Dynamics: The Basics with Applications</i> – John D. Anderson, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-07-001685-2</li> <li><i>Computational Fluid Dynamics</i> volume 1 – Klaus A. Hoffmann, Steve T. Chiang, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-9623731-0-9</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="https://www.thermalfluidscentral.org/encyclopedia/index.php/">One-Dimensional_Steady-State_Convection_and_Diffusion#Central_Difference_Scheme</a></li> <li><a href="http://www.iue.tuwien.ac.at/phd/heinzl/node27.html">Finite Differences</a></li> <li><a href="http://www.phy.davidson.edu/fachome/dmb/py200/centraldiff.htm">Central Difference Methods</a> <a href="https://web.archive.org/web/20131105095541/http://www.phy.davidson.edu/fachome/dmb/py200/centraldiff.htm">Archived</a> 5 November 2013 at the <a href="/facts/Wayback_Machine/nmQ3a6JC">Wayback Machine</a></li> <li><a href="https://arxiv.org/abs/1303.3769/">A Conservative Finite Difference Scheme for Poisson–Nernst–Planck Equations</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Computational fluid dynamics –T CHUNG, ISBN 0-521-59416-2 <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>An introduction to computational fluid dynamics by HK VERSTEEG and W. MALALASEKERA, ISBN 0-582-21884-5 <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>An introduction to computational fluid dynamics by HK VERSTEEG and W. MALALASEKERA, ISBN 0-582-21884-5 <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Liu, Xu-Dong; Tadmor, Eitan (1998). "Third order nonoscillatory central scheme for hyperbolic conservation laws". Numerische Mathematik. 79 (3): 397–425. CiteSeerX 10.1.1.26.4631. doi:10.1007/s002110050345. S2CID 16702600. <a href="/wiki/CiteSeerX_(identifier)" target="_blank">/wiki/CiteSeerX_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Liu, Xu-Dong; Tadmor, Eitan (1998). "Third order nonoscillatory central scheme for hyperbolic conservation laws". Numerische Mathematik. 79 (3): 397–425. CiteSeerX 10.1.1.26.4631. doi:10.1007/s002110050345. S2CID 16702600. <a href="/wiki/CiteSeerX_(identifier)" target="_blank">/wiki/CiteSeerX_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Lecture 5 - Solution Methods <a href="http://www.bakker.org/dartmouth06/engs150/05-solv.ppt" target="_blank">http://www.bakker.org/dartmouth06/engs150/05-solv.ppt</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> </ol>