10.3 Spectral Stability Criterion for Finite-Difference Cauchy Problems

Transcription

1 Finite-Difference Schees for Partial Differential Equations Spectral Stability Criterion for Finite-Difference Cauchy Probles Perhaps the ost widely used approach to the analysis of stability for finitedifference Cauchy probles has been proposed by von Neann. In this section, we will introduce and illustrate it using several exaples of difference equations with constant coefficients. The case of variable coefficients will be addressed in Section 1.4 and the case of initial boundary value probles (as opposed to pure initial value probles) will be explored in Section Stability with Respect to Initial Data The siplest exaple of a finite-difference Cauchy proble is the first order upwind schee: L h u (h) = f (h), (1.69) for which the operator L h and the right-hand side f (h) are given by: p+1 p up +1 up, =,±1,±2,..., L h u (h) h = p =,1,...,[T /] 1, u, =,±1,±2,..., { f (h) ϕ, p =,±1,±2,..., p =,1,...,[T /] 1, = ψ, =,±1,±2,... (1.7) We have already encountered this schee on any occasions. Let us define the nors u (h) Uh and f (h) Fh as axi nors (alternatively called l or C): u (h) def Uh = axsup u p p, f (h) def Fh = ax p sup Then for the schee (1.69) (1.7), the stability condition (1.17): ϕ p + sup ψ. (1.71) given by Definition 1.2 transfors into: [ sup c p ax p sup u (h) Uh c f (h) Fh ] ϕ p + sup ψ, p =,1,...,[T /], (1.72) where the constant c is not supposed to depend on h (or on = rh, r = const). Condition (1.72) ust hold for any arbitrary {ψ } and {ϕ p }. In particular, it should obviously hold for an arbitrary {ψ } and {ϕ p }. In other words, for

2 35 A Theoretical Introduction to Nerical Analysis stability it is necessary that solution u p to the proble: p+1 p up +1 up =, =,±1,±2,..., p =,1,...,[T /] 1, h u = ψ, =,±1,±2,..., (1.73) satisfy the inequality: sup p csup ψ, p =,1,...,[T /], (1.74) for any bounded grid function ψ. Property (1.74), which is necessary for the finite-difference schee (1.69) (1.7) to be stable in the sense (1.72), is called stability with respect to perturbations of the initial data, or siply stability with respect to the initial data. It eans that if a perturbation is introduced into the initial data ψ of proble (1.73), then the corresponding perturbation of the solution will be no ore than c ties greater in agnitude than the original perturbation of the data, where the constant c does not depend on the grid size h A Necessary Spectral Condition for Stability For the Cauchy proble (1.69) (1.7) to be stable with respect to the initial data it is necessary, in particular, that inequality (1.74) hold for ψ being equal to a haronic function: u = ψ = e iα, =,±1,±2,..., (1.75) where α is a real paraeter. One can easily solve proble (1.73) with the initial condition (1.75); the solution u p can be found in the for: u p = λ p e iα, =,±1,±2,..., p =,1,...,[T /], (1.76) where the quantity λ = λ (α) is deterined by substitution of expression (1.76) into the hoogeneous finite-difference equation of proble (1.73): Solution (1.76) satisfies the equality: sup λ (α)=1 r + re iα, r = /h = const. (1.77) u p = λ (α) p sup u = λ (α) p sup ψ. Therefore, for the stability condition (1.74) to be true it is necessary that the following inequality hold for all real α: λ (α) p c, p =,1,...,[T/].

3 Finite-Difference Schees for Partial Differential Equations 351 Equivalently, we can require that λ (α) 1 + c 1, (1.78) where c 1 is a constant that does not depend either on α or on. Inequality (1.78) represents the necessary spectral condition for stability due to von Neann. It is called spectral because of the following reason. Existence of the solution in the for (1.76) shows that the haronic e iα is an eigenfunction of the transition operator fro tie level t p to tie level t p+1 : p+1 =(1 r) p + rup +1, =,±1,±2,... According to the finite-difference equation (1.73), this operator aps the grid function {}, p =,±1,±2,...,defined for t = t p onto the grid function { p+1 }, =,±1,±2,...,defined for t = t p+1. The quantity λ (α) given by forula (1.77) is therefore an eigenvalue of the transition operator that corresponds to the eigenfunction {e iα }. In the literature, λ (α) is soeties also referred to as the aplification factor, we have encountered this concept in Section The set of all coplex nbers λ = λ (α) obtained when the paraeter α sweeps through the real axis fors a curve on the coplex plane. This curve is called the spectr of the transition operator. Consequently, the necessary stability condition (1.78) can be re-forulated as follows: The spectr of the transition operator that corresponds to the difference equation of proble (1.73) ust belong to the disk of radius 1+c 1 centered at the origin on the coplex plane. In our particular exaple, the spectr (1.77) does not depend on at all. Therefore, condition (1.78) is equivalent to the requireent that the spectr λ = λ (α) belong to the unit disk: λ (α) 1. (1.79) Let us now use the criterion that we have forulated, and actually analyze stability of proble (1.69) (1.7). The spectr (1.77) fors a circle of radius r centered at the point (1 r,) on the coplex plane. When r < 1, this circle lies inside the unit disk, being tangent to the unit circle at the point λ = 1. When r = 1 the spectr coincides with the unit circle. Lastly, when r > 1 the spectr lies outside the unit disk, except one point λ = 1, see Figure 1.5. Accordingly, the necessary stability condition (1.79) is satisfied for r 1 and is violated for r > 1. Let us now recall that in Section we have studied the sae difference proble and have proven that it is stable when r 1 and is unstable when r > 1. Therefore, in this particular case the von Neann necessary stability condition appears sufficiently sensitive to distinguish between the actual stability and instability. In the case of general Cauchy probles for finite-difference equations and systes, we give the following DEFINITION 1.3 The spectr of a finite-difference proble is given by the set of all those and only those λ = λ (α,h), for which the corresponding

4 352 A Theoretical Introduction to Nerical Analysis I λ I λ r r 1 r 1 Re λ 1 r 1 Re λ (a) r < 1 (b) r > 1 FIGURE 1.5: Spectra of the transition operators for the upwind schee. hoogeneous finite-difference equation or syste has a solution of the for: p = λ p [ u e iα], =,±1,±2,..., p =,1,...,[T/], (1.8) where u is either a fixed nber in the case of one scalar difference equation (this nber ay be taken equal to one with no loss of generality), or a constant finite-diensional vector in the case of a syste of difference equations. Then, the von Neann necessary stability condition says that given an arbitrarily sall nber ε >, for all sufficiently sall grid sizes h the spectr λ = λ (α,h) of the difference proble has to lie inside the following disk on the coplex plane: λ 1 + ε. (1.81) Note that if for a particular proble the spectr appears to be independent of the grid size h (and ), then condition (1.81) becoes equivalent to the requireent that the spectr λ (α,h)=λ(α) belong to the unit disk, see (1.79). If the von Neann necessary condition (1.81) is not satisfied, then one should not expect stability for any reasonable choice of nors. If, on the other hand, this condition is et, then one ay hope that for a certain appropriate choice of nors the schee will turn out stable Exaples We will exploit the von Neann spectral condition to analyze stability of a nberofinterestingfinite-difference probles. First, we will consider the schees that approxiate the Cauchy proble: u t u = ϕ(x,t), < x <, < t T, x (1.82) u(x,)=ψ(x), < x <.

5 Finite-Difference Schees for Partial Differential Equations 353 Exaple 1 Consider the first order downwind schee: p+1 p up u p 1 = ϕ p, h =,±1,±2,..., p =,1,...,[T /] 1, u = ψ, =,±1,±2,... Substituting the solution of type (1.76) into the corresponding hoogeneous finite-difference equation, we obtain: λ (α)=1 + r re iα. Therefore, the spectr is a circle of radius r centered at the point (1+r,) on the coplex plane, see Figure 1.6. This spectr does not depend on h. It is also clear that for no value of r does it belong I λ 1 r 1+r Re λ FIGURE 1.6: Spectr for the downwind schee. to the unit circle. Consequently, the stability condition (1.79) ay never be satisfied. This conclusion is expected, because the downwind schee obviously violates the Courant, Friedrichs, and Lewy condition for any r = /h (see Section 1.1.4). Exaple 2 Next, consider the Lax-Wendroff schee: p+1 p up +1 up 1 2h 2h 2 (up +1 2up + up 1 )=ϕ p, u = ψ, (1.83) that approxiates proble (1.82) with second order accuracy if ϕ, and with first order accuracy otherwise. For this schee, the spectr λ = λ (α,h) is deterined fro the equation: λ 1 eiα e iα 2h 2h 2 (eiα 2 + e iα )=. Let us denote r = /h as before, and notice that e iα e iα = sinα, 2i ( ) 2 e iα 2 + e iα e iα/2 e iα/2 = = sin 2 α 4 2i 2.

6 354 A Theoretical Introduction to Nerical Analysis Then, λ (α)=1 + ir sinα 2r 2 sin 2 α 2, ( λ (α) 2 = 1 2r 2 sin 2 α ) 2 + r 2 sin 2 α. 2 The spectr does not depend on h, and fro the previous equality we easily find that 1 λ 2 = 4r 2 (1 r 2 )sin 4 α 2. The von Neann condition is satisfied when the right-hand side of this equality is non-negative, which eans r 1; it is violated when r > 1. Exaple 3 Consider the explicit central-difference schee: p+1 p up +1 up 1 = ϕ p 2h, =,±1,±2,..., p =,1,...,[T /] 1, u = ψ, =,±1,±2,..., (1.84) for the sae Cauchy proble (1.82). I λ 1+i/h 1 Re λ 1 i /h FIGURE 1.7: Spectr for schee (1.84). Substituting expression (1.76) into the hoogeneous counterpart of the difference equation fro (1.84), we have: λ 1 eiα e iα =, 2h which yields: ( ) λ (α)=1 + i h sinα. The spectr λ = λ (α) fills the vertical interval of length 2/h that crosses through the point (1,) on the coplex plane, see Figure 1.7. If /h = r = const, then the spectr can be said to be independent of h (and of ). This spectr lies outside of the unit disk, the von Neann condition (1.79) is not et, and the schee is unstable. However, if we require that the teporal size be proportional to h 2 as h, so that r = /h 2 = const, then the point of the spectr, which is ost distant fro the origin, will be λ (π/2) [and also λ ( π/2)]. For this pointwe have: ( ) 2 r λ (π/2) = 1 + = 1 + r < 1 + h 2.

7 Finite-Difference Schees for Partial Differential Equations 355 Then, the von Neann condition (1.81) in the for λ (α) 1 + c 1 is satisfied for c 1 = r/2. It is clear that the requireent = rh 2 puts a considerably stricter constraint on the rate of decay of the teporal grid size as h than the previous requireent = rh does. Still, that previous requireent was sufficient for the von Neann condition to hold for the difference schees (1.69) (1.7) and (1.83) that approxiate the sae Cauchy proble (1.82). We also note that the Courant, Friedrichs, and Lewy condition of Section allows us to clai that the schee (1.84) is unstable only for /h = r > 1, but does not allow us to judge the stability for /h = r 1. As such, it appears weaker than the von Neann condition. Exaple 4 The instability of schee (1.84) for /h = r = const can be fixed by changing the way the tie derivative is approxiated. Instead of (1.84), consider the schee: p+1 (u p 1 + up +1 )/2 up +1 up 1 = ϕ p 2h, u = ψ, (1.85) obtained by replacing p with (u p 1 + up +1 )/2. This is, in fact, a general approach attributed to Friedrichs, and the schee (1.85) is known as the Lax-Friedrichs schee; we have first introduced it in Section The equation to deterine the spectr for the schee (1.85) reads: λ (e iα + e iα )/2 eiα e iα 2h which yields: λ cosα isinα = h and λ (α)=cosα + ir sinα, where r = /h = const. Consequently, λ (α) 2 = cos 2 α + r 2 sin 2 α. =, Clearly, the von Neann condition (1.79) is satisfied for r 1, because then λ 2 cos 2 α + sin 2 α = 1. For r > 1, the von Neann condition is violated. Exaple 5 Finally, consider the leap-frog schee (1.36) for proble (1.82). The corresponding hoogeneous finite-difference equation is written as: u p+1 p 1 2 up +1 up 1 2h =, (1.86)

8 356 A Theoretical Introduction to Nerical Analysis and for the spectr we obtain: λ λ 1 eiα e iα =. 2 2h This is a quadratic equation with respect to λ : λ 2 i2rλ sinα 1 =, where r = /h = const. The roots of this equation are given by: λ 1,2 = ir sin α ± 1 r 2 sin 2 α. We notice that when r 1, then λ 1,2 2 = r 2 sin 2 α +(1 r 2 sin 2 α)=1, so that the entire spectr lies precisely on the unit circle and the von Neann condition (1.79) is satisfied. Otherwise, when r > 1, we again take α = π/2 and obtain: λ 1 = ir + i r 2 1 = r + r 2 1 > 1, which eans that the von Neann condition is not et. We thus see that the von Neann criterion can be applied to a finitedifference equation that connects ore than two consecutive tie levels on the grid. However, for r = 1 the two-step schee (1.86) proves unstable, see Section Next, we will consider two schees that approxiate the following Cauchy proble for the heat equation: u t 2 u a2 = ϕ(x,t), x2 < x <, < t T, u(x,)=ψ(x), < x <. (1.87) Exaple 6 The first schee is explicit: p+1 p a 2 up +1 2up + u p 1 h 2 = ϕ p, u = ψ, =,±1,±2,..., p =,1,...,[T/] 1. It allows us to copute { p+1 } in the closed for via {}: p u p+1 =(1 2ra 2 )u p + ra 2 (u p +1 + up 1 )+ϕp, p =,1,...,[T /] 1, where r = /h 2 = const. Substitution of u p = λ p e iα into the corresponding hoogeneous difference equation yields: By noticing that λ 1 a 2 e iα 2 + e iα h 2 =. e iα 2 + e iα = sin 2 α 4 2,

9 Finite-Difference Schees for Partial Differential Equations 357 we obtain: λ (α)=1 4ra 2 sin 2 α 2, r = h 2. When α varies between and 2π, the point λ (α) sweeps the interval 1 4ra 2 λ 1 of the real axis, see Figure 1.8. For stability, it is necessary that the left endpoint of this interval still be inside the unit circle (Figure 1.8), i.e., that I λ 1 4ra 2 1. This requireent translates into: r 1 1 4ra 2 2a 2. (1.88) 1 Re λ Inequality (1.88) guarantees that the von Neann stability condition will hold. Conversely, if we have r > 1/(2a 2 ),then the point λ (α)=1 4ra 2 sin 2 (α/2) that corresponds to α = π will be located to the left of the point 1, i.e., outside the FIGURE 1.8: Spectr of the explicit schee for the heat equation. unit circle. In this case, the haronic e iπ =( 1) generates the solution: u p =(1 4ra2 ) p ( 1) that does not satisfy inequality (1.74) for any value of c. Exaple 7 The second schee is iplicit: p+1 p a 2 up up+1 + u p+1 1 h 2 = ϕ p+1, u = ψ, =,±1,±2,..., p =,1,...,[T/] 1. For this schee, { p+1 } cannot be obtained fro {} p by an explicit forula because for a given the difference equation contains three, rather than one, unknowns: u p+1 +1, up+1,andu p+1 1. As in the previous case, we substitute up = λ p e iα into the hoogeneous equation and obtain: λ (α)= ra 2 sin 2 (α/2), r = h 2. Consequently, the spectr of the schee fills the interval: ( 1 + 4ra 2 ) 1 λ 1 of the real axis and the von Neann condition λ 1isetforanyr.

10 358 A Theoretical Introduction to Nerical Analysis Exaple 8 The von Neann analysis also applies when studying stability of a schee in the case of ore than one spatial variable. Consider, for instance, a Cauchy proble for the heat equation on the (x,y) plane: u t = 2 u x u y 2 < x, y <, < t T, u(x,y,)=ψ(x,y), < x, y <. We approxiate this proble on the unifor Cartesian grid: (x,y n,t p )= (h,nh, p). Replacing the derivatives with difference quotients we obtain:,n p+1,n p = up +1,n 2up,n + u p 1,n h 2 + u p,n+1 2up,n + u p,n 1 h 2 u,n = ψ,n,,n =,±1,±2,..., p =,1,...,[T /] 1. The resulting schee is explicit. To analyze its stability, we specify u,n in the for of a two-diensional haronic e i(α+β n) deterined by two real paraeters α and β, and generate a solution in the for: u p,n = λ p (α,β )e i(α+β n). Substituting this expression into the hoogeneous difference equation, we find after soe easy equivalence transforations: ( λ (α,β )=1 4r sin 2 α 2 + β ) sin2. 2 When the real quantities α and β vary between and 2π, the point λ = λ (α,β ) sweeps the interval 1 8r λ 1 of the real axis. The von Neann stability condition is satisfied if 1 8r 1, i.e., whenr 1/4. Exaple 9 In addition to the previous Exaple 5, let us now consider another exaple of the schee that connects the values of the difference solution on the three, rather than two, consecutive tie levels of the grid. A Cauchy proble for the one-diensional hoogeneous d Alebert (wave) equation: 2 u t 2 2 u =, < x <, < t T, x2 u(x,)=ψ () u(x,) (x), = ψ (1) (x), < x <, t

11 Finite-Difference Schees for Partial Differential Equations 359 can be approxiated on a unifor Cartesian grid by the following schee: u p+1 2u p + u p 1 2 up +1 2up + u p 1 h 2 =, u = ψ () u 1, u = ψ (1), h =,±1,±2,..., p = 1,2,...,[T /] 1. Substituting a solution of type (1.76) into the foregoing finite-difference equation, we obtain the following quadratic equation for deterining λ = λ (α): ( λ r 2 sin 2 α ) λ + 1 =, r = 2 h. The product of the two roots of this equation is equal to one. If its discriinant: D(α)=4r 2 sin 2 α (r 2 sin 2 α ) is negative, then the roots λ 1 (α) and λ 2 (α) are coplex conjugate and both have a unit odulus. If r < 1, the discriinant D(α) reains negative for all α [, 2π). I λ I λ 1 1 Re λ 1 1 Re λ (a) r < 1 (b) r > 1 FIGURE 1.9: Spectr of the schee for the wave equation. The spectr for this case is shown in Figure 1.9(a); it fills an arc of the unit circle. If r = 1, the spectr fills exactly the entire unit circle. When r > 1, the discriinant D(α) ay be either negative or positive depending on the value of α. In this case, once the argent α increases fro to π the roots λ 1 (α) and λ 2 (α) depart fro the point λ = 1 and ove along the unit circle: One root oves clockwise and the other root counterclockwise, and then they erge at λ = 1. After that one root oves away fro this point along the real axis to the right, and the other one to the left, because they are both real and their product λ 1 λ 2 = 1, see Figure 1.9(b). The von Neann stability condition is et for r 1.

12 36 A Theoretical Introduction to Nerical Analysis Consider a Cauchy proble for the following first order hyperbolic syste of equations that describes the propagation of acoustic waves: v t = w x, w = v t x, < x <, < t T, v(x,)=ψ (1) (x), w(x,)=ψ (2) (x). (1.89a) Let us set: u(x,t)= [ ] v(x,t), ψ(x)= w(x, t) Then, proble (1.89a) can be recast in the atrix for: [ ψ (1) ] (x) ψ (2). (x) u t A u =, < x <, < t T, x u(x,)=ψ(x), < x <, (1.89b) where [ ] 1 A =. 1 We will now analyze two difference schees that approxiate proble (1.89b). Exaple 1 Consider the schee: p+1 p A up +1 up =, u = ψ. (1.9) h We will be looking for a solution to the vector finite-difference equation (1.9) in the for (1.8): [ ] p = λ p v w e iα. Substituting this expression into equation (1.9) we obtain: or alternatively, λ 1 u A eiα 1 u =, h (λ 1)u r(e iα 1)Au =, r = /h. (1.91) Equality (1.91) can be considered as a vector for of the syste of linear algebraic equations with respect to the coponents of the vector u. Syste (1.91) can be written as: [ λ 1 r(e iα ][ ] [ 1) v r(e iα =. (1.92) 1) λ 1 ] w

13 Finite-Difference Schees for Partial Differential Equations 361 [ ] v Syste (1.92) ay only have a nontrivial solution u = if its deterinant turns into zero, which yields the following equation for λ = λ (α): Consequently, λ 1 (α)=1 r + re iα, λ 2 (α)=1 + r re iα. When the paraeter α varies between and 2π, the roots λ 1 (α) and λ 1 (α) sweep two circles of radius r centered at the locations 1 r and 1 + r, respectively. Therefore, the von Neann stability conditions ay never be satisfied; irrespective of any particular value of r. (λ 1) 2 = r 2 (e iα 1) 2. 1 Exaple 11 Consider the vector Lax-Wendroff schee: I λ 1 r w 1+r 1 Re λ FIGURE 1.1: Spectr of schee (1.9). p+1 p A up +1 up 1 2h 2h 2 A2 (u p +1 2up + up 1 )=, u = ψ, =,±1,±2,..., p =,1,...,[T /] 1, (1.93) that approxiates proble (1.89b) on its sooth solutions with second order accuracy and that is analogous to the scalar schee (1.83) for the Cauchy proble (1.82). As in Exaple 1, the finite-difference equation of (1.93) ay only have a non-trivial solution of[ type ](1.8) if the deterinant of the corresponding linear v syste for finding u = w turns into zero. Writing down this deterinant and requiring that it be equal to zero, we obtain a quadratic equation with respect to λ = λ (α). Its roots can be easily found: λ 1 (α)=1 + ir sinα 2r 2 sin 2 α 2, λ 2 (α)=1 ir sinα 2r 2 sin 2 α 2. These forulae are analogous to those obtained for the scalar Exaple 2, and siilarly to that exaple we have: 1 λ 1,2 (α) 2 = 4r 2 sin 4 α 2 (1 r2 ). We therefore see that the spectr {λ 1 (α),λ 2 (α)} belongs to the unit disk if r 1.

14 362 A Theoretical Introduction to Nerical Analysis Stability in C Let us ephasize that the type of stability we have analyzed in Sections was stability in the sense of the axi nor (1.71). Alternatively, it is referred to as stability in (the space) C. This space contains all bounded nerical sequences. The von Neann spectral condition (1.78) is necessary for the schee to be stable in C. As far as the sufficient conditions, in soe siple cases stability in C can be proved directly, for exaple, using axi principle, as done in Section for the first order explicit upwind schee and in Section for an explicit schee for the heat equation. Otherwise, sufficient conditions for stability in C turn out to be delicate and ay require rather sophisticated argents. The analysis of a general case even for one scalar constant coefficient difference equation goes beyond the scope of the current book, and we refer the reader to the original work by Fedoryuk [Fed67] (see also his onograph [Fed77, Chapter V, 4]). In addition, in [RM67, Chapter 5] the reader can find an account of the work by Strang and by Thoee on the subject Sufficiency of the Spectral Stability Condition in l 2 However, a sufficient condition for stability ay soeties be easier to find if we were to use a different nor instead of the axi nor (1.71). Let, for exaple, u p 2 = = u (h) Uh = ax p u p 2, ϕ p 2 = = ϕ p 2, ψ 2 = = ψ 2, u p, f (h) Fh = ϕ p ψ = ax{ ψ, ax ϕ p }. p Fh (1.94) Relations (1.94) define Euclidean (i.e., l 2 ) nors for u p, ϕ p,andψ. Accordingly, stability in the sense of the nors given by (1.94) is referred to as stability in (the space) l 2. We recall that the space l 2 is a Hilbert space of all nerical sequences, for which the s of squares of absolute values of all their ters is bounded. Consider a general constant coefficient finite-difference Cauchy proble: under the assption that j right j right b j u p+1 + j a j u p + j = ϕ p, (1.95) j= j left j= j left u = ψ, =,±1,±2,..., p =,1,...,[T/] 1, j right j= j left b j e iα j, α < 2π. Note that all spatially one-diensional schees fro Section fit into the category (1.95), even for a relatively narrow range: j left = j right = 1.

15 Finite-Difference Schees for Partial Differential Equations 363 THEOREM 1.3 For the schee (1.95) to be stable in l 2 with respect to the initial data, i.e., for the following inequality to hold: u p c ψ, p =,1,...,[T /], (1.96) where the constant c does not depend on h [or on = (h)], is is necessary and sufficient that the von Neann condition (1.78) be satisfied, i.e., that the spectr of the schee λ = λ (α) belong to the disk: λ (α) 1 + c 1, (1.97) where c 1 is another constant that does not depend either on α or on. PROOF We will first prove the sufficiency. By hypotheses of the theore, the nber series ψ 2 converges. Then, the function series of the = independent variable α: ψ e iα = also converges in the space L 2 [,2π], and its s that we denote Ψ(α), α 2π, is a function that has ψ as the Fourier coefficients: ψ = 1 2π Ψ(α)e iα dα, 2π =,±1,±2,..., (1.98) (a realization of the Riesz-Fischer theore, see, e.g., [KF75, Section 16]). Moreover, the following relation holds: ψ 2 = = ψ 2 = 1 2π Ψ(α) 2 dα = 1 2π 2π Ψ 2 2 known as the Parseval equality. Consider a hoogeneous counterpart to the difference equation (1.95): j right j right b j u p+1 + j a j u p + j =, j= j left j= j left =,±1,±2,..., p =,1,...,[T /] 1, For any α [,2π) this equation obviously has a solution of the type: u p (α)=λ p (α)e iα (1.99)

16 364 A Theoretical Introduction to Nerical Analysis for soe particular λ = λ (α) that can be deterined by substitution: λ (α)= Then, the grid function: ( jright )( jright ) 1 a j e iα j b j e iα j. j= j left j= j left p = 1 2π Ψ(α)λ p (α)e iα dα, 2π =,±1,±2,..., (1.1) provides solution to the Cauchy proble (1.95) for the case ϕ p = because it is a linear cobination of solutions (α) p of (1.99), and coincides with ψ for p =, see (1.98). Note that the integral on the right-hand side of forula (1.1) converges by virtue of the Parseval equality, because 2π Ψ(α) 2 dα < = 2π Ψ(α) dα <. If the von Neann spectral condition (1.97) is satisfied, then λ (α) p 1 + c 1 T / e c 1T. (1.11) Consequently, using representation (1.1) along with the Parseval equality and inequality (1.11), we can obtain: u p 2 = 1 2π λ p (α)ψ(α) 2 dα e 2c 1T 1 2π 2π 2π Ψ(α) 2 dα = e 2c 1T ψ 2, which clearly iplies stability with respect to the initial data: u p c ψ. To prove the necessity, we will need to show that if (1.97) holds for no fixed c 1, then the schee is unstable. We should ephasize that to deonstrate the instability for the chosen nor (1.94) we ay not exploit the unboundedness of the solution u p = λ p (α)e iα that takes place in this case, because the grid function {e iα } does not belong to l 2. Rather, let us take a particular Ψ(α) L 2 [,2π] such that 2π 1 2π λ (α) 2p Ψ(α) 2 dα ax α ( λ (α) 2p ε ) 2π 1 Ψ(α) 2 dα, (1.12) 2π where ε > is given. For an arbitrary ε, estiate (1.12) can always be guaranteed by selecting: { 1, if α [α δ,α + δ], Ψ(α)=, if α [α δ,α + δ], where α = argax α λ (α) and δ >. Indeed, as the function λ (α) 2p is continuous, inequality (1.12) will hold for a sufficiently sall δ = δ(ε).

17 Finite-Difference Schees for Partial Differential Equations 365 If estiate (1.11) does not take place, then we can find a sequence h k, k =,1,2,3,..., and the corresponding sequence k = (h k ) such that ( c k = ax α ) [T λ (α,h /k ] k) as k. Let us set ε = 1 and choose Ψ(α) to satisfy (1.12). Define ψ as Fourier coefficients of the function Ψ(α), according to forula (1.98). Then, inequality (1.12) for p k =[T/ k ] transfors into: u p k 2 (c 2 k 1) ψ 2 = u p k (c k 1) ψ, c k as k, i.e., there is indeed no stability (1.96) with respect to the initial data. Theore 1.3 establishes equivalence between the von Neann spectral condition and the l 2 stability of schee (1.95) with respect to the initial data. In fact, one can go even further and prove that the von Neann spectral condition is necessary and sufficient for the full-fledged l 2 stability of the schee (1.95) as well, i.e., when the right-hand side ϕ p is not disregarded. One iplication, the necessity, iediately follows fro Theore 1.3, because if the von Neann condition does not hold, then the schee is unstable even with respect to the initial data. The proof of the other iplication, the sufficiency, can be found in [GR87, 25]. This proof is based on using the discrete Green s functions. In general, once stability with respect to the initial data has been established, stability of the full inhoogeneous proble can be derived using the Duhael principle. This principle basically says that the solution to the inhoogeneous proble can be obtained as linear superposition of the solutions to soe specially chosen hoogeneous probles. Consequently, a stability estiate for the inhoogeneous proble can be obtained on the basis of stability estiates for a series of hoogeneous probles, see [Str4, Chapter 9] Scalar Equations vs. Systes As of yet, our analysis of finite-difference stability has focused ostly on scalar equations; we have considered a 2 2 syste only in Exaples 1 and 11 of Section In Exaples 5 and 9, we have also considered scalar difference equations that connect the values of the solution on ore than two consecutive tie levels; those can be reduced to systes on two tie levels. In general, a constant coefficient finite-difference Cauchy proble with vector unknowns (i.e., a syste) can be written in the for siilar to (1.95): j right j right B j u p+1 + j A j u p + j = ϕ p, (1.13) j= j left j= j left u = ψ, =,±1,±2,..., p =,1,...,[T /] 1,

18 366 A Theoretical Introduction to Nerical Analysis under the assption that the atrices j right B j e iα j, α < 2π, j= j left are non-singular. In forula (1.13),, p ϕ p,andψ are grid vector functions of a fixed diension, and A j = A j (h), B j = B j (h), j = j left,..., j right, are given square atrices of the sae diension. Solution to the hoogeneous counterpart of equation (1.13) can be sought for in the for (1.8), where u = u (α,h) and λ = λ (α,h) are the eigenvectors and eigenvalues, respectively, of the aplification atrix of schee (1.13): Λ = Λ(α,h)= ( jright ) 1 ( jright ) B j e iα j j= j left A j e iα j j= j left (1.14) The von Neann spectral condition (1.97) is clearly necessary for stability of finite-difference systes in all nors. Indeed, if it is not et, then estiate (1.11) will not hold, and the schee will develop a catastrophic exponential instability that cannot be fixed by any reasonable choice of nors. Yet the von Neann condition reains only a necessary stability condition for systes in either C or l 2.ForC, the analysis of sufficient conditions becoes cbersoe already for the scalar case (see Section 1.3.4), and even in l 2 obtaining sufficient conditions for systes proves rather involved. Qualitatively, the difficulties ste fro the fact that the aplification atrix (1.14) ay have ultiple eigenvalues and as a consequence, ay not necessarily have a full set of eigenvectors. If a ultiple eigenvalue occurs exactly on the unit circle or just outside the unit disk, this ay still cause instability even when all the eigenvalues satisfy the von Neann constraint (1.97) (siilar to Theore 9.2). An exaple is provided by the leap-frog schee (1.86). If r = 1andα = π/2, then we have λ 1,2 = i, andifr = 1andα = 3π/2, then λ 1,2 = i. In either case, in addition to (1.8) there will be a solution of the for u p = pλ p [ u e iα],whichis a anifestation of a gradually (linearly) developing instability. Of course, if the aplification atrix appears noral (a atrix that coutes with its adjoint) and therefore unitarily diagonalizable, then none of the aforeentioned difficulties is present, and the von Neann condition becoes not only necessary but also sufficient for stability of the vector schee (1.13) in l 2. Otherwise, the question of stability for schee (1.13) can be equivalently reforulated using the new concept of stability for failies of atrices. A faily of square atrices (of a given fixed diension) is said to be stable if there is a constant K > such that for any particular atrix Λ fro the faily, and any positive integer p, the following estiate holds: Λ p K. Schee (1.13) is stable in l 2 if and only if the faily of aplification atrices Λ = Λ(α,h) given by (1.14) is stable in the sense of the previous definition (this faily is paraeterized by α [, 2π) and h > ). Theore 1.4, known as the Kreiss atrix theore, provides soe necessary and sufficient conditions for a faily of atrices to be stable.

19 Finite-Difference Schees for Partial Differential Equations 367 THEOREM 1.4 (Kreiss) Stability of a faily of atrices Λ is equivalent to any of the following: 1. There is a constant C 1 > such that for any atrix Λ fro the given faily, and any coplex nber z, z > 1, there is a resolvent (Λ zi) 1 bounded as: (Λ zi) 1 C 1 z There are constants C 2 > and C 3 >, and for any atrix Λ fro the given faily there is a non-singular atrix M such that M C 2, M 1 C 2,andtheatrixD def = MΛM 1 is upper triangular, with the off-diagonal entries that satisfy: d ij C 3 in{1 κ i,1 κ j }, where κ i = d ii and κ j = d jj are the corresponding diagonal entries of D, i.e., the eigenvalues of Λ. 3. There is a constant C 4 >, and for any atrix Λ fro the given faily there is a Heritian positive definite atrix H, such that C 1 4 I H C 4I and Λ HΛ H. The proof can be found in [RM67, Chapter 4] or [Str4, Chapter 9]. Exercises 1. Consider the so-called weighted schee for the heat equation: p+1 p = σ up up+1 + u p+1 1 h 2 +(1 σ) up +1 2up + u p 1 h 2, u = ψ, =,±1,±2,..., p =,1,...,[T /] 1, where the real paraeter σ [,1] is called the weight (between the fully explicit schee, σ =, and fully iplicit schee, σ = 1). What values of σ guarantee that the schee will eet the von Neann stability condition for any r = /h 2 = const? 2. Consider the Cauchy proble (1.87) for the heat equation. The schee: u p+1 p 1 2 a 2 up +1 2up + u p 1 h 2 = ϕ p, u = ψ, u 1 = ψ, =,±1,±2,..., p =,1,...,[T /] 1, whereweassethatϕ(x, ) and define: ψ = u(x,)+ u(x,) = u(x,)+a 2 2 u(x,) x=x t x=x x 2 = ψ + a 2 ψ (x ), approxiates proble (1.87) on its sooth solutions with accuracy O( 2 +h 2 ).Does this schee satisfy the von Neann spectral stability condition for r = /h 2 = const?

20 368 A Theoretical Introduction to Nerical Analysis 3. For the two-diensional Cauchy proble: u t u x u = ϕ(x,y,t), y < x, y <, < t T, u(x,y,)=ψ(x,y), < x, y <, investigate the von Neann spectral stability of: a) The first order explicit schee:,n p+1,n p up +1,n up,n up,n+1 up,n = ϕ p h h,n, u,n = ψ,n,, n =,±1,±2,..., p =,1,...,[T/] 1; b) The second order explicit schee:,n p+1,n p 1 up +1,n up 1,n up,n+1 up,n 1 = ϕ,n p 2 2h 2h, u,n = ψ,n, u 1,n = ψ,n + [ψ x(x,y n )+ψ y(x,y n )+ϕ(x,y n,)],, n =,±1,±2,..., p =,1,...,[T/] Investigate the von Neann spectral stability of the iplicit two-diensional schee for the hoogeneous heat equation:,n p+1,n p = up+1 +1,n 2up+1,n + u p+1 1,n h 2 + u p+1,n+1 2up+1,n + u p+1,n 1 h 2 u,n = ψ,n,,n =,±1,±2,..., p =,1,...,[T /] Investigate the von Neann stability of the iplicit upwind schee for the Cauchy proble (1.82): p+1 p up+1 +1 up+1 = ϕ p h, u = ψ, =,±1,±2,..., p =,1,...,[T /] 1. (1.15) 6. Investigate the von Neann stability of the iplicit downwind schee for the Cauchy proble (1.82): p+1 p up+1 u p+1 1 = ϕ p h, u = ψ, =,±1,±2,..., p =,1,...,[T /] 1. (1.16) 7. Investigate the von Neann stability of the iplicit central schee for the Cauchy proble (1.82): p+1 p up+1 +1 up+1 1 = ϕ p 2h, u = ψ, =,±1,±2,..., p =,1,...,[T /] 1. (1.17)

21 Finite-Difference Schees for Partial Differential Equations Transfor the leap-frog schee (1.86) of Exaple 5, Section 1.3.3, and the centraldifference schee for the d Alebert equation of Exaple 9, Section 1.3.3, to the schees written for finite-difference systes, as opposed to scalar equations, but connecting only two, as opposed to three, consecutive tie levels of the grid. Investigate the von Neann stability by calculating spectra of the corresponding aplification atrices (1.14). Hint. Use the difference { p+1 } p as the second unknown grid function. 1.4 Stability for Probles with Variable Coefficients The von Neann necessary condition that we have introduced in Section 1.3 to analyze stability of linear finite-difference Cauchy probles with constant coefficients can, in fact, be applied to a wider class of forulations. A siple extension that we describe in this section allows one to exploit the von Neann condition to analyze stability of probles with variable coefficients (continuous, but not necessarily constant) and even soe nonlinear probles The Principle of Frozen Coefficients Introduce a unifor Cartesian grid: x = h, =,±1,±2,..., t p = p, p =,1,2,..., and consider a finite-difference Cauchy proble for the hoogeneous heat equation with the variable coefficient of heat conduction a = a(x,t): p+1 p a(x,t p ) up +1 2up + u p 1 h 2 =, u = ψ(x ), =,±1,±2,..., p. (1.18) Next, take an arbitrary point ( x, t) in the doain of proble (1.18) and freeze the coefficients of proble (1.18) at this point. Then, we arrive at the constantcoefficient finite-difference equation: p+1 p a( x, t) up +1 2up + u p 1 h 2 =, =,±1,±2,..., p. (1.19) Having obtained equation (1.19), we can forulate the following principle of frozen coefficients. For the original variable-coefficient Cauchy proble (1.18) to be stable it is necessary that the constant-coefficient Cauchy proble for the difference equation (1.19) satisfy the von Neann spectral stability condition. To justify the principle of frozen coefficients, we will provide an heuristic argent rather than a proof. When the grid is refined, the variation of the coefficient a(x,t) in a neighborhood of the point ( x, t) becoes saller if easured over any