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Upwind scheme
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<h2 id="model-equation">Model equation</h2> <p>To illustrate the method, consider the following one-dimensional linear <a href="/facts/Advection_equation/3rn50V2r">advection equation</a> </p> ∂ u ∂ t + a ∂ u ∂ x = 0 {\displaystyle {\frac {\partial u}{\partial t}}+a{\frac {\partial u}{\partial x}}=0} <p>which describes a wave propagating along the x {\displaystyle x} -axis with a velocity a {\displaystyle a} . This equation is also a mathematical model for one-dimensional linear <a href="/facts/Advection/3rn50V2r">advection</a>. Consider a typical grid point i {\displaystyle i} in the domain. In a one-dimensional domain, there are only two directions associated with point i {\displaystyle i} – left (towards negative infinity) and right (towards positive infinity). If a {\displaystyle a} is positive, the traveling wave solution of the equation above propagates towards the right, the left side is called the <i>upwind</i> side and the right side is the <i>downwind</i> side. Similarly, if a {\displaystyle a} is negative the traveling wave solution propagates towards the left, the left side is called <i>downwind</i> side and right side is the <i>upwind</i> side. If the finite difference scheme for the spatial derivative, ∂ u / ∂ x {\displaystyle \partial u/\partial x} contains more points in the upwind side, the scheme is called an upwind-biased or simply an upwind scheme. </p> <h2 id="first-order-upwind-scheme">First-order upwind scheme</h2> <p>The simplest upwind scheme possible is the first-order upwind scheme. It is given by<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p> <table><tbody><tr><td> u i n + 1 − u i n Δ t + a u i n − u i − 1 n Δ x = 0 for a > 0 {\displaystyle {\frac {u_{i}^{n+1}-u_{i}^{n}}{\Delta t}}+a{\frac {u_{i}^{n}-u_{i-1}^{n}}{\Delta x}}=0\quad {\text{for}}\quad a>0} </td> <td></td> <td>1</td></tr></tbody></table> <table><tbody><tr><td> u i n + 1 − u i n Δ t + a u i + 1 n − u i n Δ x = 0 for a < 0 {\displaystyle {\frac {u_{i}^{n+1}-u_{i}^{n}}{\Delta t}}+a{\frac {u_{i+1}^{n}-u_{i}^{n}}{\Delta x}}=0\quad {\text{for}}\quad a<0} </td> <td></td> <td>2</td></tr></tbody></table> <p>where n {\displaystyle n} refers to the t {\displaystyle t} dimension and i {\displaystyle i} refers to the x {\displaystyle x} dimension. (By comparison, a central difference scheme in this scenario would look like </p> u i n + 1 − u i n Δ t + a u i + 1 n − u i − 1 n 2 Δ x = 0 , {\displaystyle {\frac {u_{i}^{n+1}-u_{i}^{n}}{\Delta t}}+a{\frac {u_{i+1}^{n}-u_{i-1}^{n}}{2\Delta x}}=0,} <p>regardless of the sign of a {\displaystyle a} .) </p> <h3>Compact form</h3> <p>Defining </p> a + = max ( a , 0 ) , a − = min ( a , 0 ) {\displaystyle a^{+}={\text{max}}(a,0)\,,\qquad a^{-}={\text{min}}(a,0)} <p>and </p> u x − = u i n − u i − 1 n Δ x , u x + = u i + 1 n − u i n Δ x {\displaystyle u_{x}^{-}={\frac {u_{i}^{n}-u_{i-1}^{n}}{\Delta x}}\,,\qquad u_{x}^{+}={\frac {u_{i+1}^{n}-u_{i}^{n}}{\Delta x}}} <p>the two conditional equations (1) and (2) can be combined and written in a compact form as </p> <table><tbody><tr><td> u i n + 1 = u i n − Δ t [ a + u x − + a − u x + ] {\displaystyle u_{i}^{n+1}=u_{i}^{n}-\Delta t\left[a^{+}u_{x}^{-}+a^{-}u_{x}^{+}\right]} </td> <td></td> <td>3</td></tr></tbody></table> <p>Equation (3) is a general way of writing any upwind-type schemes. </p> <h3>Stability</h3> <p>The upwind scheme is <a href="/facts/Numerical_stability/jRDdo8zV">stable</a> if the following <a href="/facts/Courant%25E2%2580%2593Friedrichs%25E2%2580%2593Lewy_condition/f4r00Wca">Courant–Friedrichs–Lewy condition</a> (CFL) is satisfied.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> c = | a Δ t Δ x | ≤ 1 {\displaystyle c=\left|{\frac {a\Delta t}{\Delta x}}\right|\leq 1} and 0 ≤ a {\displaystyle 0\leq a} . <p>A <a href="/facts/Taylor_series/M0aTuftH">Taylor series</a> analysis of the upwind scheme discussed above will show that it is first-order accurate in space and time. Modified wavenumber analysis shows that the first-order upwind scheme introduces severe <a href="/facts/Numerical_diffusion/P1yncze2">numerical diffusion</a>/dissipation in the solution where large gradients exist due to necessity of high wavenumbers to represent sharp gradients. </p> <h2 id="second-order-upwind-scheme">Second-order upwind scheme</h2> <p>The spatial accuracy of the first-order upwind scheme can be improved by including 3 data points instead of just 2, which offers a more accurate finite difference stencil for the approximation of spatial derivative. For the second-order upwind scheme, u x − {\displaystyle u_{x}^{-}} becomes the 3-point backward difference in equation (3) and is defined as </p> u x − = 3 u i n − 4 u i − 1 n + u i − 2 n 2 Δ x {\displaystyle u_{x}^{-}={\frac {3u_{i}^{n}-4u_{i-1}^{n}+u_{i-2}^{n}}{2\Delta x}}} <p>and u x + {\displaystyle u_{x}^{+}} is the 3-point forward difference, defined as </p> u x + = − u i + 2 n + 4 u i + 1 n − 3 u i n 2 Δ x {\displaystyle u_{x}^{+}={\frac {-u_{i+2}^{n}+4u_{i+1}^{n}-3u_{i}^{n}}{2\Delta x}}} <p>This scheme is less diffusive compared to the first-order accurate scheme and is called linear upwind differencing (LUD) scheme. </p> <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/Upwind_differencing_scheme_for_convection/j2jo7gIB">Upwind differencing scheme for convection</a></li> <li><a href="/facts/Godunov%2527s_scheme/l4YaCvoG">Godunov's scheme</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Courant, Richard; Isaacson, E; Rees, M. (1952). "On the Solution of Nonlinear Hyperbolic Differential Equations by Finite Differences". Comm. Pure Appl. Math. 5 (3): 243..255. doi:10.1002/cpa.3160050303. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Patankar, S. V. (1980). Numerical Heat Transfer and Fluid Flow. Taylor & Francis. ISBN 978-0-89116-522-4. <a href="978-0-89116-522-4" target="_blank">978-0-89116-522-4</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Hirsch, C. (1990). Numerical Computation of Internal and External Flows. John Wiley & Sons. ISBN 978-0-471-92452-4. <a href="978-0-471-92452-4" target="_blank">978-0-471-92452-4</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> </ol>