... differencing of equation (19.1 .20 ) is n+1 n rj+1 /2 − rj+1 /2 n+1 /2 sj n+1 /2 = ∆t n−1 /2 − sj =v ∆t n+1 /2 − sj ∆x n n rj+1 /2 − rj−1 /2 sj+1 (19.1.35) ∆x If you substitute equation (19.1 .22 ) in equation (19.1.35), ... shows that the amplification factor for equation (19 .2. 4) is ξ =1− 4D∆t sin2 (∆x )2 k∆x (19 .2. 5) The requirement |ξ| ≤ leads to the stability criterion 2D∆t ≤1 (∆x )2 (19 .2. 6) Sample page from NUMERICAL ... values uj+1 /2 at the half timesteps tn+1 /2 and the half mesh points xj+1 /2 These are calculated by the Lax scheme: n+1 /2 uj+1 /2 = n ∆t (u (F n − Fjn ) + un ) − j j+1 2 x j+1 (19.1.37) n+1 /2 Using...
... 19.0 .2) , uj+1,l − 2uj,l + uj−1,l uj,l+1 − 2uj,l + uj,l−1 + = ρj,l 22 (19.0.5) or equivalently uj+1,l + uj−1,l + uj,l+1 + uj,l−1 − 4uj,l = 2 ρj,l (19.0.6) To write this system of linear equations ... vector F is called the conserved flux For example, the prototypical hyperbolic equation, the one-dimensional wave equation with constant velocity of propagation v ∂2u ∂2u = v2 ∂t2 ∂x (19.1 .2) ... Numerical Methods for PartialDifferential Equations, 2nd ed (New York: Academic Press) [1] Richtmyer, R.D., and Morton, K.W 1967, Difference Methods for Initial Value Problems, 2nd ed (New York:...
... going to see all hell break loose! 19.3 Initial Value Problems in Multidimensions 855 The expression for |ξ |2 can be manipulated into the form |ξ |2 = − (sin2 kx∆ + sin2 ky ∆) − ( 2 + 2 ) x y ... The last two terms are negative, and so the stability requirement |ξ |2 ≤ becomes − ( 2 + 2 ) ≥ x y ∆ ∆t ≤ √ 2 2(vx + vy )1 /2 or (19.3.9) (19.3.10) This is an example of the general result for ... ∆ )2 − (αy sin kx ∆ − αx sin ky ∆ )2 856 Chapter 19 PartialDifferentialEquations (19.3.13) Called the alternating-direction implicit method (ADI), this embodies the powerful concept of operator...
... (19.0.6), we find umn e2πim/J + e 2 im/J + e2πin/L + e 2 in/L − = ρmn 2 or umn = ρmn 2 2πn 2 m + cos 2 cos J L (19.4.4) (19.4.5) Thus the strategy for solving equation (19.0.6) by FFT techniques is: ... ∂x 8 62 Chapter 19 PartialDifferentialEquations The finite-difference form of equation (19.4 .28 ) can be written as a set of vector equations uj−1 + T · uj + uj+1 = gj 2 (19.4 .29 ) T = B − 21 (19.4.30) ... and then adding the three equations, we get (1) uj 2 + T(1) · uj + uj +2 = gj 2 (19.4. 32) This is an equation of the same form as (19.4 .29 ), with T(1) = 21 − T2 (1) gj = 2 (gj−1 − T · gj + gj+1...
... the asymptotic formula for large J turns out to be ρs 1− 2 2J (19.5.11) The number of iterations r required to reduce the error by a factor of 10−p is thus r 2pJ ln 10 2 pJ (19.5. 12) In other ... − 2 Jacobi (19.5 .20 ) + π/J 2 1− J ω ρSOR (19.5 .21 ) for large J (19.5 .22 ) Equation (19.5.10) gives for the number of iterations to reduce the initial error by a factor of 10−p , r pJ ln 10 2 ... Chapter 19 PartialDifferentialEquations The Gauss-Seidel method, equation (19.5.6), corresponds to the matrix decomposition (L + D) · x(r) = −U · x(r−1) + b (19.5.13) ρs r 1− 2 J2 pJ ln 10 2 (19.5.14)...
... + h2 u2 (x, y) + · · · Then, assuming R is of high enough order that we can neglect its effect, equation (19.6. 32) gives τh LH (u + h2 u2 ) − Lh (u + h2 u2 ) = LH (u) − Lh (u) + h2 [LH (u2 ) ... Problems 881 u [2] [2] = -h*h*rhs [2] [2] /4.0; } h=1.0/(n-1); h2=h*h; for (ipass=1;ipass< =2; ipass++,jsw=3-jsw) { Red and black sweeps isw=jsw; for (j =2; j
... Universitext For other titles in this series, go to www.springer.com/series /22 3 Haim Brezis Functional Analysis, Sobolev Spaces and PartialDifferentialEquations 1C Haim Brezis Distinguished Professor ... interesting application to PDEs, due toY Pinchover, is presented in S Agmon [2] For a proof of the Krein–Milman theorem, see Problem (b) In the theory of partialdifferentialequations Let us mention, ... E; x2n = ∀n ≥ 1} and Y = y = (yn )n≥1 ∈ E; y2n = y2n−1 ∀n ≥ 2n Check that X and Y are closed linear spaces and that X + Y = E Let c ∈ E be defined by 24 The Hahn–Banach Theorems Introduction to...
... We want to show jh 1; T 2ij C k 1k2k 2k2 for all 1; 2 S (Rn): We note that h 1; Pt2TPt2 2i ! as t ! and hence it is enough to prove Z d j dt h 1; Pt2TPt2 2idtj C k 1k2k 2k2 for all 1; 2 S (Rn) ... it is enough to show that Z j h 1; RtQtTPt2 2i dt j C k 1k2k 2k2 for all 1; 2 S (Rn): t Hence it is enough to show Z j h 1; RtQtTPt2 2i dt j C (k 1k2 + k 2k2) for all 1; 2 S (Rn): 2 t But Z Z ... 1; QtTPt2 2ij dt t 0Z dt + Z kQ TP k2 dt I + II: krt 1k2 t t t 2 t 0 33 Here I= Z ^ k ' t(jtjj j) ^1( )k2 dt C k ^1k2 = C k 1k2: t 2To cope with II, we rewirte QtTPt2 = LtPt as ((LtPt) 2) (x) =...
... then (a) j(x y)j2 (x x) (y y) x y2V (b) kxk (x x)1 =2 de nes a norm k k on V for which kx + yk2 + kx ; yk2 = 2( kxk2 + kyk2 ) x y2V (c) the scalar product is continuous from V and V to K Proof : ... A show \fM : Ag V 2. 1 Prove parts (d) and (e) of Lemma 2. 1 2.2 If V1 p1 and V2 p2 are seminormed spaces, show p(x) p1 (x1 ) + p2(x2 ) is a seminorm on the product V1 V2 2. 3 Let V p be a seminormed ... r1 ) Similarly, there is an integer n2 > n1 and s(y2 r2 ) s(y1 r1 ) such that r2 < 1 =2 UNIFORM BOUNDEDNESS WEAK COMPACTNESS 23 and j(xn2 y)H j > for y s(y2 r2 ) We inductively de ne s(yj rj ) s(yj;1...