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CHAPTER 2. PARABOLIC EQUATIONS 2.3 The Convection Diffusion Equation The ConvectionDiﬀusion Equation: ut + aux = buxx If a = 0, we have the heat equation, and if b = 0, we have the wave equation. Deﬁne a new function w such that w(t, x − at) = u(t, x), i.e., w(t, x) = u(t, x + at). Then bwxx = buxx = ut + aux = wt − awx + awx = wt . So u is simply the solution to the heat equation translated with speed a. The problem occurs when the viscosity coeﬃcient b is very small compared to a. Then the obvious numerical methods trick you. We have a choice of n+1 vm − vm n v n − vm−1 n n n v n − 2vm + vm−1 + a m+1 = b m+1 , (2.2) k 2h h2 n+1 n n n n n n vm − vm v − vm v − 2vm + vm−1 + a m+1 = b m+1 , k h h2 n+1 vm − vm n n v n − vm−1 n n v n − 2vm + vm−1 +a m = b m+1 . (2.3) k h h2 We expand in Taylor series to see that the orders of accuracy are O(k + h2 ), O(k + h), and O(k + h), respectively, which speaks strongly in favor of (2.2). The heat equation has one nice property. The maximum at a later time is less than the maximum at an earlier time. We will copy that. Rewrite (2.2) as � � � � n+1 aλ n n aλ n vm = bµ − vm+1 + (1 − 2bµ)vm + bµ + vm−1 2 2 � � � � aλ n n aλ n = bµ 1 − vm+1 + (1 − 2bµ)vm + bµ 1 + vm−1 . 2bµ 2bµ aλ ah aλ Let α ≡ 2bµ = 2b . Now if all the coeﬃcients are positive (i.e., |α| = | 2bµ | < 1), then n+1 n n n |vm | ≤ bµ(1 − α) max |vm | + (1 − 2bµ) max |vm | + bµ(1 + α) max |vm | m m m n ≤ [bµ(1 − α) + (1 − 2bµ) + bµ(1 + α)] max |vm | m n ≤ max |vm |. m The maximum is a decreasing function of time if |α| ≤ 1, i.e., if |a| h 2b · ≤ 1, which is the same as h≤ . (2.4) b 2 |a| This will, of course, be satisﬁed eventually as h → 0, but who can wait that long? Say a = 10, b = 10−2 ⇒ h ≈ 10−3 is needed. And remember, for stability we must have 1/2 ≥ bµ = 10−2 · k/(10−3 )2 . This implies k ≈ 10−4 /2, which is terribly small. Now look at (2.3) instead: n+1 n n n n n n vm − vm + aλ(vm − vm−1 ) = bµ(vm+1 − 2vm + vm−1 ), which we rewrite as n+1 n n n vm = (bµ + aλ)vm−1 + (1 − 2bµ − aλ)vm + bµvm+1 aλ n aλ = bµ(1 + )vm−1 + (1 − 2bµ(1 + n ))v n + bµvm+1 bµ 2bµ m n n n = bµ(1 + 2α)vm−1 + (1 − 2bµ(1 + α))vm + bµvm+1 . 2.4. SUMMARY OF SCHEMES FOR THE HEAT EQUATION Say a > 0, so α > 0. Now the requirement for maxnorm stability becomes 2bµ(1 + α) < 1 or 2bµ + aλ < 1, which is a lot less restrictive than (2.4). We can pick h to be 10−2 instead of 10−3 , i.e., 10 times larger. We −3 try k = 105 , h = 10−2 , a = 10, b = 10−2 . Then −3 10 k k 2 1 2b 2 + a = 2 · 10−2 · 5 −2 )2 = + < 1. h h (10 50 5 So we increased the timestep by a factor of 10. But at what price? We can rewrite (2.3) as n+1 n n n � n n n vm+1 − vm−1 ah vm+1 − 2vm + vm−1 � vm − vm +a = b+ . k 2h 2 h2 ah We introduced an artiﬁcial viscosity 2 = bα. This artiﬁcial viscosity comes from our numerical method. Therefore solving ut + aux = b(1 + α)uxx using (2.2) is equivalent to solving ut + aux = buxx using (2.3).

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convection-diffusion equation, convection-diﬀusion equation, diffusion equation, heat equation, dx dy, time step, j. math, diffusion equations, ﬁnite element, boundary condition, variational principle, fundamental solution, initial data, upper bound, convection-diﬀusion equations

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posted: | 4/22/2010 |

language: | English |

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