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Lax–Wendroff method
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<h2 id="definition">Definition</h2> <p>Suppose one has an equation of the following form: ∂ u ( x , t ) ∂ t + ∂ f ( u ( x , t ) ) ∂ x = 0 {\displaystyle {\frac {\partial u(x,t)}{\partial t}}+{\frac {\partial f(u(x,t))}{\partial x}}=0} where x and t are independent variables, and the initial state, <i>u</i>(<i>x</i>, 0) is given. </p> <h3>Linear case</h3> <p>In the linear case, where <i>f</i>(<i>u</i>) = <i>Au</i>, and <i>A</i> is a constant,<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> u i n + 1 = u i n − Δ t 2 Δ x A [ u i + 1 n − u i − 1 n ] + Δ t 2 2 Δ x 2 A 2 [ u i + 1 n − 2 u i n + u i − 1 n ] . {\displaystyle u_{i}^{n+1}=u_{i}^{n}-{\frac {\Delta t}{2\Delta x}}A\left[u_{i+1}^{n}-u_{i-1}^{n}\right]+{\frac {\Delta t^{2}}{2\Delta x^{2}}}A^{2}\left[u_{i+1}^{n}-2u_{i}^{n}+u_{i-1}^{n}\right].} Here n {\displaystyle n} refers to the t {\displaystyle t} dimension and i {\displaystyle i} refers to the x {\displaystyle x} dimension. This linear scheme can be extended to the general non-linear case in different ways. One of them is letting A ( u ) = f ′ ( u ) = ∂ f ∂ u {\displaystyle A(u)=f'(u)={\frac {\partial f}{\partial u}}} </p> <h3>Non-linear case</h3> <p>The conservative form of Lax-Wendroff for a general non-linear equation is then: u i n + 1 = u i n − Δ t 2 Δ x [ f ( u i + 1 n ) − f ( u i − 1 n ) ] + Δ t 2 2 Δ x 2 [ A i + 1 / 2 ( f ( u i + 1 n ) − f ( u i n ) ) − A i − 1 / 2 ( f ( u i n ) − f ( u i − 1 n ) ) ] . {\displaystyle u_{i}^{n+1}=u_{i}^{n}-{\frac {\Delta t}{2\Delta x}}\left[f(u_{i+1}^{n})-f(u_{i-1}^{n})\right]+{\frac {\Delta t^{2}}{2\Delta x^{2}}}\left[A_{i+1/2}\left(f(u_{i+1}^{n})-f(u_{i}^{n})\right)-A_{i-1/2}\left(f(u_{i}^{n})-f(u_{i-1}^{n})\right)\right].} where A i ± 1 / 2 {\displaystyle A_{i\pm 1/2}} is the Jacobian matrix evaluated at 1 2 ( u i n + u i ± 1 n ) {\textstyle {\frac {1}{2}}(u_{i}^{n}+u_{i\pm 1}^{n})} . </p> <h2 id="jacobian-free-methods">Jacobian free methods</h2> <p>To avoid the Jacobian evaluation, use a two-step procedure. </p> <h3>Richtmyer method</h3> <p>What follows is the Richtmyer two-step Lax–Wendroff method. The first step in the Richtmyer two-step Lax–Wendroff method calculates values for <i>f</i>(<i>u</i>(<i>x</i>, <i>t</i>)) at half time steps, <i>t</i><i>n</i> + 1/2 and half grid points, <i>x</i><i>i</i> + 1/2. In the second step values at <i>t</i><i>n</i> + 1 are calculated using the data for <i>t</i><i>n</i> and <i>t</i><i>n</i> + 1/2. </p><p>First (Lax) steps: u i + 1 / 2 n + 1 / 2 = 1 2 ( u i + 1 n + u i n ) − Δ t 2 Δ x ( f ( u i + 1 n ) − f ( u i n ) ) , {\displaystyle u_{i+1/2}^{n+1/2}={\frac {1}{2}}(u_{i+1}^{n}+u_{i}^{n})-{\frac {\Delta t}{2\,\Delta x}}(f(u_{i+1}^{n})-f(u_{i}^{n})),} u i − 1 / 2 n + 1 / 2 = 1 2 ( u i n + u i − 1 n ) − Δ t 2 Δ x ( f ( u i n ) − f ( u i − 1 n ) ) . {\displaystyle u_{i-1/2}^{n+1/2}={\frac {1}{2}}(u_{i}^{n}+u_{i-1}^{n})-{\frac {\Delta t}{2\,\Delta x}}(f(u_{i}^{n})-f(u_{i-1}^{n})).} </p><p>Second step: u i n + 1 = u i n − Δ t Δ x [ f ( u i + 1 / 2 n + 1 / 2 ) − f ( u i − 1 / 2 n + 1 / 2 ) ] . {\displaystyle u_{i}^{n+1}=u_{i}^{n}-{\frac {\Delta t}{\Delta x}}\left[f(u_{i+1/2}^{n+1/2})-f(u_{i-1/2}^{n+1/2})\right].} </p> <h3>MacCormack method</h3> <p class="note">Main article: <a href="/facts/MacCormack_method/rKcMRJBa">MacCormack method</a></p> <p>Another method of this same type was proposed by MacCormack. MacCormack's method uses first forward differencing and then backward differencing: </p><p>First step: u i ∗ = u i n − Δ t Δ x ( f ( u i + 1 n ) − f ( u i n ) ) . {\displaystyle u_{i}^{*}=u_{i}^{n}-{\frac {\Delta t}{\Delta x}}(f(u_{i+1}^{n})-f(u_{i}^{n})).} Second step: u i n + 1 = 1 2 ( u i n + u i ∗ ) − Δ t 2 Δ x [ f ( u i ∗ ) − f ( u i − 1 ∗ ) ] . {\displaystyle u_{i}^{n+1}={\frac {1}{2}}(u_{i}^{n}+u_{i}^{*})-{\frac {\Delta t}{2\Delta x}}\left[f(u_{i}^{*})-f(u_{i-1}^{*})\right].} </p><p>Alternatively, First step: u i ∗ = u i n − Δ t Δ x ( f ( u i n ) − f ( u i − 1 n ) ) . {\displaystyle u_{i}^{*}=u_{i}^{n}-{\frac {\Delta t}{\Delta x}}(f(u_{i}^{n})-f(u_{i-1}^{n})).} Second step: u i n + 1 = 1 2 ( u i n + u i ∗ ) − Δ t 2 Δ x [ f ( u i + 1 ∗ ) − f ( u i ∗ ) ] . {\displaystyle u_{i}^{n+1}={\frac {1}{2}}(u_{i}^{n}+u_{i}^{*})-{\frac {\Delta t}{2\Delta x}}\left[f(u_{i+1}^{*})-f(u_{i}^{*})\right].} </p> <ul><li>Michael J. Thompson, <i>An Introduction to Astrophysical Fluid Dynamics</i>, Imperial College Press, London, 2006.</li> <li>Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007). <a href="http://apps.nrbook.com/empanel/index.html#pg=1040">"Section 20.1. Flux Conservative Initial Value Problems"</a>. <i>Numerical Recipes: The Art of Scientific Computing</i> (3rd ed.). New York: Cambridge University Press. p. 1040. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-88068-8.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>P.D Lax; B. Wendroff (1960). "Systems of conservation laws" (PDF). Commun. Pure Appl. Math. 13 (2): 217–237. doi:10.1002/cpa.3160130205. Archived from the original on September 25, 2017. <a href="https://apps.dtic.mil/sti/pdfs/ADA385056.pdf" target="_blank">https://apps.dtic.mil/sti/pdfs/ADA385056.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>LeVeque, Randall J. (1992). Numerical Methods for Conservation Laws (PDF). Boston: Birkhäuser. p. 125. ISBN 0-8176-2723-5. <a href="0-8176-2723-5" target="_blank">0-8176-2723-5</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> </ol>