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Part IX, Chapter 53 Parabolic PDEs: time approximation We are concerned in this chapter with both the time and the space approximation of the model problem (52.4). We adopt the method of line introduced in §52.2.1 and approximate the semi-discrete problem (52.13) wit respect to time. In the entire chapter we keep the same notation and conventions as in Chapter 52. 53.1 Implicit Euler We introduce the backward Euler method to approximate the semi-discrete problem (52.13) and investigate the stability and convergence properties of the resulting algorithm. For this section and the rest of the chapter we set T where N is a nonzero natural number. ∆t = N 53.1.1 Principle Let u0h ∈ Vh be a reasonable approximation of u0 . We formalize this construction by assuming that there is an operator Ih ∈ L(L; Vh ) such that u0h := Ih u0 . We moreover asume that (Ih )h>0 is uniformly bounded in L(L; L). For instance we can take Ih = PhL , where PhL is the L-orthogonal projection from L to Vh , i.e., (PhL (v), vh )L = (v, vh )L for all vh ∈ Vh and all v ∈ L. We will abuse the notation by denoting again PhL the extension of PhL to V ′ , i.e., (PhL f, vh )L = hf, vh iV ′ ,V . Note that this make sense since Vh ⊂ V and Vh is finite-dimensional, i.e., sup06=vh ∈Vh kvh kV /kvh kL < ∞. The implicit Euler (or backward Euler method) consists of constructing an n approximating sequence uh∆t = (u0h , . . . , unh , . . . uN h ), uh ∈ Vh , n ∈ {0:N }, by mean of the following time-stepping algorithm: ∀vh ∈ Vh , (un+1 − unh , vh )L h +a(tn+1 , un+1 , vh ) = (PhL f (tn+1 ), vh )L , (53.1) h ∆t 742 Chapter 53. Parabolic PDEs: time approximation where we expect un+1 to be a reasonable approximation of u(tn+1 ), where h n+1 n t = t + ∆t = (n + 1)∆t, 0 ≤ n ≤ N − 1. To analyze the properties of this algorithm, we introduce the operator N Sh∆t : Vh ×VhN −→ VhN +1 such that, for xh ∈ Vh and yh∆t ∈ VhN , the sequence N zh∆t = (zh0 , . . . , zhN ) = Sh∆t (xh , yh∆t ) is inductively defined by zh0 = xh and ∀vh ∈ Vh , (zhn+1 − zhn , vh )L + a(tn+1 , zhn+1 , vh ) = (yhn+1 , vh )L . ∆t (53.2) Upon setting ρh∆t (f ) = (PhL f (t1 ), . . . , PhL f (tN )), we realize that (53.1) can be re-written as follows: N uh∆t = Sh∆t (Ih (u0 ), ρh∆t (f )). Note that we abuse the notation and indifferently write zh∆t = (zh0 , . . . , zhN ) or zh∆t = (zh1 , . . . , zhN ) when the context is unambiguous. Remark 53.1. Note that we assumed that f ∈ C 0 ((0, T ]; V ′ ) for (53.1) to make sense. This assumption is made in the entire chapter. It can be lifted in the context of the backward Euler method by replacing f (tn+1 ) by R tn+1 1 f (τ ) dτ . ⊓ ⊔ ∆t tn 53.1.2 Stability We define kyh kVh′ := sup06=zh ∈Vh (yh , zh )L /kzh kV . Let n0 , n1 ∈ {0:N } and let B be a normed space; we additionally consider the following discrete norms: kyh∆t kℓ2 ([tn0 ,tn1 ];B) := kyh∆t kℓ∞ ([tn0 ,tn1 ];B) := ∆t n=n X1 n=n0 kyhn k2B max kyhn kB . n0 ≤n≤n1 ! 12 , (53.3) (53.4) N The operator Sh∆t has stability properties that are similar to those of the continuous problem stated in Theorem 52.12 (i.e., continuous dependence with respect to the data). N (xh , yh∆t ) Lemma 53.2 (Stability). Assume ∆t ≤ αc12 and let zh∆t = Sh∆t P be defined in (53.2). Then, the following holds for any n ∈ {0:N }: ( 1 2 n e− 4 αcP t kxh kL + √1α kyh∆t kℓ2 ([t1 ,tn ];Vh′ ) , n kzh kL ≤ (53.5) 2 n 1 e− 4 αcP t kxh kL + αc1P kyh∆t kℓ∞ ([t1 ,tn ];Vh′ ) , ( 1 √ kxh kL + 1 kyh∆t kℓ2 ([t1 ,T ];V ′ ) , α α h √ (53.6) kzh∆t kℓ2 ([t1 ,tN ];V ) ≤ √1 kxh kL + T kyh∆t kℓ∞ ([t1 ,T ];V ′ ) . α α h Part IX. Time-Dependent PDEs 743 Proof. (1) We test (53.2) with 2∆tzhn+1 . The relation 2p(p − q) = p2 + (p − q)2 − q 2 and the coercivity of a imply that kzhn+1 k2L + kzhn+1− zhn k2L + 2∆tαkzhn+1 k2V ≤ kzhn k2L + 2∆tkyhn+1 kVh′ kzhn+1 kV ≤ kzhn k2L + n+1 2 ∆t kV ′ α kyh h + ∆tαkzhn+1 k2V . The last inequality is obtained from the arithmetic–geometric inequality (A.14). The inequality cP kzhn+1 kL ≤ kzhn+1 kV leads to n+1 2 (1+∆tαc2P )kzhn+1 k2L ≤ kzhn+1 k2L+ ∆tαkzhn+1 k2V ≤ kzhn k2L+ ∆t kV ′ . (53.7) α kyh h The Discrete Gronwall Lemma (see Lemma 53.3 below), in turn, implies that n kzhn+1 k2L ≤ kxh k2L ∆t X 1 + ky k+1 k2V ′ . (53.8) h (1 + ∆tαc2P )n+1 α (1 + ∆tαc2P )n−k+1 h k=0 ∆tαc2P )−1 − 12 ∆tαc2P Note that (1 + ≤ e if ∆tαc2P ≤ 1. Moreover, (1 + 2 −1 ∆tαcP ) < 1. The first estimate in (53.5) follows easily. The second estimate follows from (53.8) by using the inequality n X k=0 1 1 ≤ . 2 n−k+1 (1 + ∆tαcP ) ∆tαc2P (2) Summing (53.7) from n = 0 to N − 1 leads to ∆tα N X n=1 kzhn k2V ≤ kxh k2L N X 1 ky n k2 ′ , + ∆t α n=1 h Vh ⊓ ⊔ whence the estimates in (53.6) are readily deduced. Lemma 53.3 (Discrete Gronwall). Let γ 6= −1 and β be two real numbers. Let (an )n≥0 , (f n )n≥0 be two sequences such that (1+γ)an+1 ≤ an +βf n . Then, n an+1 ≤ X a0 fk + β . (1 + γ)n+1 (1 + γ)n−k+1 k=0 Remark 53.4. N (i) Since uh∆t = Sh∆t (Ih (u0 ), (PhL f )∆t ), Lemma 53.2, together with the ′ bound cP kf kV ≤ kf kL , implies that the solution to (53.1) satisfies the following stability estimates: 1 √1 T 2 n kunh kL ≤ e− 4 αcP t ku0 kL + kuh∆t kℓ2 ([t1 ,T ];V ) ≤ √1 αT ku0 kL + 1 αcP 1 αc2P kf kC 0 ([0,tn ];L) , kf kC 0 ([0,T ];L) . (ii) The assumption ∆t ≤ αc12 is just made to make the estimates (53.5)P (53.6) look nice; it is not a stability restriction on the time step. The algorithm is said to unconditionally stable. ⊓ ⊔ 744 Chapter 53. Parabolic PDEs: time approximation 53.1.3 Error analysis Theorem 53.5. Let u be the solution to (52.4). Assuming that u ∈ C 2 [0, T ]; V ′ )∩ C 1 ([0, T ]; L) we have 2 1 n ku(tn ) − unh kL ≤ e− 4 αcP t ke0h kL + kηh kC 0 ([0,tn ];L) 1 αc2P + √1 T keh∆t kℓ2 ([t1 ,tN ];V ) ≤ √1 αT 1 αcP ∆t|u|C 2 ([0,tn ];V ′ ) ke0h kL + kηh kC 0 ([0,T ];L) 1 αcP + |ηh |C 1 ([0,tn ];L) + |ηh |C 1 ([0,T ];L) + α1 ∆t|u|C 2 ([0,T ];V ′ ) . Proof. We are going to proceed like in the proof of Theorem 52.19. (1) Let Pht be the elliptic projector defined in (52.15), let wh (t) := Pht (u) for any t ∈ [0, T ], and denote whn := wh (tn ) for any n ∈ {0:N }. By definition, wh0 = Ph0 (u). The equation (52.13) can be re-written as follows: n+1 n (wh −wh ,vh )L ∆t +a(tn+1 , whn+1 , vh ) = (f (tn+1 ), vh )L +(PhL Rn+1 , vh )L , (53.9) R tn+1 R tn+1 1 1 where Rn+1 := ∆t (dτ wh (τ ) − dτ u(τ )) dτ − ∆t (τ − tn )dτ τ u(τ ) dτ. Let tn tn n n n n n n us now introduce eh := wh − uh and ηh = u(t ) − wh . Subtracting (53.1) from (53.9), we obtain that (en+1 −en h ,vh )L h ∆t + a(tn+1 , en+1 , vh ) = (PhL Rn+1 , vh )L , h ∀vh ∈ Vh . Setting Rh∆t = (PhL R1 , . . . , PhL RN ), the above argument shows that N eh∆t = Sh∆t (e0h , Rh∆t ). We conclude by invoking the stability Lemma 53.2: 2 1 √1 T n kenh kL ≤ e− 4 αcP t ke0h kL + keh∆t kℓ2 ([t1 ,tN ];V ) ≤ √1 αT 1 αcP kRh∆t kℓ∞ ([0,tn ];Vh′ ) , ke0h kL + α1 kRh∆t kℓ∞ ([0,T ];Vh′ ) . n+1 We (roughly) estimate kRh∆t kVh′ as follows: n+1 kRh∆t kVh′ ≤ 1 cP |ηh |C 1 ([tn ,tn+1 ];L) + ∆t|u|C 2 ([t, tn+1 ];V ′ ) , which immediately implies that 1 2 n ku(tn ) − unh kL ≤ e− 4 αcP t ke0h kL + kηh kC 0 ([0,tn ];L) + √1 T keh∆t kℓ2 ([t1 ,tN ];V ) ≤ 1 αc2P √1 αT + The conclusion follows readily. |ηh |C 1 ([0,tn ];L) + 1 αcP ∆t|u|C 2 ([0,tn ];V ′ ) ke0h kL + kηh kC 0 ([0,T ];L) 1 αcP |ηh |C 1 ([0,T ];L) + α1 ∆t|u|C 2 ([0,T ];V ′ ) . ⊓ ⊔ Part IX. Time-Dependent PDEs 745 Example 53.6 (Finite elements). Let us assume that Vh is composed of conforming finite elements of degree k, see §25.1.2. Let u be solution of the heat equation (or that of the equation with the bilinear form (52.12)) and asume that u ∈ C 1 ([0, T ]; H k+1 (D)) ∩ C 2 ([0, T ]; L2(D)). Assume also that the adjoint of the differential operator associated with the elliptic projector has a smoothing property in H 1+s (D), s ∈ [0, 1], (see e.g., Theorem ??), then there is c, uniform with respect to h and T , such that |u − Pht (u)|C 1 ([0,T ];L2 (D)) ≤ c hk+s |u|C 1 ([0,T ];H k+1 (D)) ku − Pht (u)kC 0 ([0,T ];H 1 (D)) ≤ c hk kukC 0([0,T ];H k+1 (D)) . Assuming that u0 is well approximated, say ke0h kL2 ≤ c hk+s , there is c uniform with respect to h, u and T such that the following error estimates hold: ku∆t − uh∆t kℓ∞ ([0,T ];L1 (D)) ≤ c(hk+s kukC 1 ([0,T ];H k+1 (D)) + ∆t|u|C 2 ([0,T ];L2 (D)) ) √1 T ku∆t − uh∆t kℓ2 ([t1 ,T ];H 1 (D)) ≤ c(hk kukC 1 ([0,T ];H k+1 (D)) + ∆t|u|C 2 ([0,T ];L2 (D)) ). Remark 53.7 (Optimality). The above error analysis is by no mean optimal. Our goal in this chapter is just to show the key mechanisms in play. ⊓ ⊔ Remark 53.8 (T → +∞). Note that the constant c is independent of T . Hence, if kukC 1 ([0,T ];H k+1 (D)) and |u|C 2 ([0,T ];L2 (D)) are bounded with respect to T , the error estimates are uniform in T , i.e., the error is uniformly controlled for arbitrary large times. This remarkable property is characteristic of parabolic equations: these equations loose memory of the initial data, and they also progressively loose memory of approximation errors made in the past, i.e., the approximation error does not accumulate as time grows. ⊓ ⊔ 53.2 Explicit Euler 53.2.1 Principle We start as in the backward Euler method by defining u0h := Ih u0 ∈ Vh to be a reasonable approximation of u0 . The forward Euler (or explicit Euler ) algorithm consists of approximating the solution to (52.4) by constructing a n+1 n sequence uh∆t = (u0h , . . . , uN ,0≤n≤ h ), uh ∈ Vh , n ∈ {0:N }, such that uh N − 1, solves the following problem: ∀vh ∈ Vh , (un+1 − unh , vh )L h + a(tn+1 , unh , vh ) = (f (tn+1 ), vh )L . ∆t (53.10) To investigate the stability of the method, we now define the operator N Sh∆t : Vh ×Vhn −→ VhN +1 such that the sequence zh∆t = (zh0 , . . . , zhN ) := 746 Chapter 53. Parabolic PDEs: time approximation N Sh∆t (xh , yh∆t ) is given by zh0 = xh and solving the following sequence of problems for all n ∈ {0:N − 1}: ∀vh ∈ Vh , n+1 1 ∆t (zh − zhn , vh )L + a(tn+1 , zhn , vh ) = (yhn+1 , vh )L . (53.11) This definition implies that, upon setting ρh∆t (f ) := (PhL f (t1 ), . . . , PhL f (tN )), N the sequence solving (53.10) satisfies uh∆t = Sh∆t (Ih (u0 ), ρh∆t (f )). 53.2.2 stability The analysis below will reveal that stability of the algorithm is controlled by the following parameter: ci (h) = max 06=vh ∈Vh kvh kV . kvh kL (53.12) This quantity is finite since Vh is finite-dimensional. If Vh is a finite element space based on a quasi-uniform mesh sequence, an inverse inequality shows that ci (h) ≤ c h−1 , see Lemma ??. Recall that there is M such that |a(t, v, w)| ≤ M kvkV kwkV for all t ∈ [0, T ], and all v, w ∈ V . Lemma 53.9 (Stability). Let θ ∈ (0, 1) be a real number. Let zh∆t = θα N Sh∆t (xh , yh∆t ) be defined in (53.11). Assume that ∆t ≤ ci (h) 2 M 2 , then 1 √1 T 2 n kzhn kL ≤ kxh kL e− 4 αcP (1−θ)t + 1 kxh kL αT (1−θ) kzh∆t kℓ2 ([t1 ,T ];V ) ≤ √ + 1 √ kyhn kℓ∞ ([0,tn ];Vh′ ) , αcP 1−θ √1 kyhn kℓ∞ ([0,T ];Vh′ ) . α (1−θ) Proof. We proceed as in the proof of Lemma 53.2. Testing (53.11) with 2∆tzhn+1 and using the relation 2p(p − q) = p2 + (p − q)2 − q 2 together with (A.14), we obtain kzhn+1 k2L +kzhn+1− zhn k2L +2∆ta(tn+1, zhn , zhn+1 ) ≤ kzhn k2L +2∆tkyhn+1 kVh′ kzhn+1 kV ≤ kzhn k2L + n+1 2 ∆t kV ′ α kyh h + ∆tαkzhn+1 k2V . Furthermore, the inverse inequality (53.12) implies a(tn+1 , zhn , zhn+1 ) = a(tn+1 , zhn+1 , zhn+1 ) + a(tn+1 , zhn − zhn+1 , zhn+1 ) ≥ αkzhn+1 k2V − M kzhn − zhn+1 kV kzhn+1 kV ≥ αkzhn+1 k2V − ci (h)M kzhn+1 − zhn kL kzhn+1 kV ≥ αkzhn+1 k2V − M 2 ci (h)2 kzhn+1 2θα − zhn k2L − n+1 2 θα kV 2 kzh As a result, kzhn+1 k2L + 1 − M 2 ci (h)2 ∆t θα kzhn+1 − zhn k2L + ∆tα(1 − θ)kzhn+1 k2V ≤ kzhn k2L + n+1 2 ∆t kV ′ . α kyh h . Part IX. Time-Dependent PDEs Since ∆t ≤ θα M 2 ci (h)2 , 747 we infer that kzhn+1 k2L + ∆tα(1 − θ)kzhn+1 k2V ≤ kzhn k2L + n+1 2 ∆t kV ′ . α kyh h ⊓ ⊔ The rest of the proof is the same as that of Lemma 53.2. Remark 53.10 (Conditional stability). The forward Euler algorithm is said to be conditionally stable because of the restriction on ∆t. ⊓ ⊔ 53.2.3 Error analysis Theorem 53.11. Let u be the solution to (52.4). Assume that u ∈ C 2 ([0, T ]; V ′ )∩ C 1 [0, T ]; V ) and ∆t < ci (h)α2 M 2 , then 1 √1 T 2 n ku(tn ) − unh kL ≤ ke0h kL e− 4 αcP (1−θ)t + ku∆t − uh∆t kℓ2 ([t1 ,T ];V ) ≤ √ where c(t, u) = 1 cP 1 ke0h kL αT (1−θ) + 1 √ c(tn , u), αcP 1−θ √1 c(T, u), α (1−θ) |ηh |C 1 ([0,t];L) + ∆t|u|C 2 ([0,t];V ′ ) + M ∆t|u|C 1 [0,t];V ) Proof. We proceed as in the proof of Theorem 53.5 . First, we investigate the consistency of the method. Let wh (t) := Pht (u) where Pht is the elliptic projector defined in (52.15) and denote whn := wh (tn ) for all n ∈ {0:N }. The definition of whn+1 implies that (52.13) can be re-written as follows: n+1 1 −whn , vh )L +a(tn+1 , whn , vh ) ∆t (wh = (f n+1 , vh )L +(Rhn+1 , vh )L , ∀vh ∈ Vh , where, Rhn+1 ∈ Vh , 0 ≤ n ≤ N − 1, is defined so that the following holds 1 (whn+1 − whn ) − ∂t u(tn+1 ), vh L −a(tn+1 , u(tn+1 )−u(tn ), vh ), (Rhn+1 , vh )L = ∆t N for all vh ∈ Vh . This definition implies that eh∆t = Sh∆t (e0h , Rh∆t ). The n+1 ′ quantity kRh kVh is (roughly) estimated by kRhn+1 kVh′ ≤ 1 cP |ηh |C 1 ([tn ,tn+1 ];L) + ∆t|u|C 2 ([t, tn+1 ];V ′ ) + M ∆t|u|C 1 [tn ,tn+1 ];V ) . The rest of the proof is the same as that of Theorem 53.5. ⊓ ⊔ Example 53.12 (Finite elements). Using the same assumptions as in Example 53.6, we infer that there is c uniform with respect to h, u and T such that the following error estimates hold if ∆t < ci (h)α2 M 2 : ku∆t − uh∆t kℓ∞ ([0,T ];L2 (D)) ≤ c(hk+s kukC 1 ([0,T ];H k+1 (D)) + ∆t|u|C 2 ([0,T ];L2 (D)) ) √1 T ku∆t − uh∆t kℓ2 ([t1 ,T ];H 1 (D)) ≤ c(hk kukC 1 ([0,T ];H k+1 (D)) + ∆t|u|C 2 ([0,T ];L2 (D)) ). 748 Chapter 53. Parabolic PDEs: time approximation 53.3 Second-order implicit schemes We present two schemes with second-order accuracy in time in this section. 53.3.1 BDF2 The so-called second-order backward Euler method is based on the approx1 (3u(tn+1 ) − 4u(tn ) − u(tn−1 )) + O(∆t2 ). This is a imation ∂t u(tn+1 ) = 2∆t two-step method, usually referred to as BDF2 in the literature, where the approximate sequence uh∆t ∈ VhN +1 is defined so that the following holds for all vh ∈ Vh and all n ∈ {0:N − 1}: 0 u = Ih u0 h 1 1 0 1 0 1 ∆t 1 2 (53.13) ∆t (uh − uh , vh )L + a( 2 , 2 (uh + uh ), vh ) = (PhL f (t ), vh )L , 1 n+1 n−1 n+1 n n+1 n+1 , vh )L + a(t , uh , vh ) = (f (t ), vh )L . 2∆t (3uh − 4uh + uh N The stability analysis can be done by defining the operator Sh∆t : N N 0 N N Vh ×Vh −→ Vh so that the sequence (zh , . . . , zh ) = Sh∆t (xh , yh∆t ) is generated according to the rule 0 zh = xh , 1 0 0 1 ∆t 1 1 1 (53.14) ∆t (zh − zh , vh )L + a( 2 , 2 (zh + zh ), vh ) = (yh , vh )L , 1 n+1 n−1 n+1 n n+1 n+1 , vh )L + a(t , zh , vh ) = (yh , vh )L , 2∆t (3zh − 4zh + zh 1 Defining ρh∆t (f ) := (PhL f (t 2 ), PhL f (t2 ), . . . , PhL f (tN )), the algorithm (53.13) N is such that uh∆t = Sh∆t (Ih u0 , ρh∆t (f )). N Lemma 53.13 (Stability). Let zh∆t = Sh∆t (xh , yh∆t ) as in (53.14). Then, 1 2 n kzhn kL ≤ c(kxh kV + ∆tkyh1 kL )e− 4 αcP t + √1 T kzh∆t kℓ2 ([t1 ,T ];V ) ≤ √c αT (kxh kV + 1 ′ ∞ 1 n αcP kyh∆t kℓ ([t ,t ];Vh ) , ∆tkyh1 kL ) + α1 kyh∆t kℓ∞ ([t1 ,T ];Vh′ ) . Proof. Test the third equation in (53.14) with 4∆tzhn+1 . Using the relation 2(an+1 , 3an+1 − 4an + an−1 ) = |an+1 |2 + |2an+1 − an |2 + |δtt an+1 |2 − |an |2 − |2an − an−1 |2 , (53.15) with δtt an+1 := an+1 − 2an + an−1 , yields kzhn+1 k2L + k2zhn+1 − zhn k2L + kδtt zhn+1 k2L + 4∆tαkzhn+1 k2V ≤ kzhn k2L + k2zhn − zhn−1 k2L + 2∆tαkzhn+1 k2V + Thus, for n ≥ 1, n+1 2 2∆t kV ′ . α kyh h Part IX. Time-Dependent PDEs 749 kzhn+1 k2L + k2zhn+1 − zhn k2L + 2∆tαkzhn+1 k2V ≤ kzhn k2L + k2zhn − zhn−1 k2L + n+1 2 2∆t kV ′ . α kyh h Furthermore, testing the second equation in (53.14) with ∆t(zh1 + zh0 ) leads to kzh1 k2L + α∆t 1 4 kzh + zh0 k2V ≤ kzh0 k2L + ∆t 1 2 α kyh kVh′ . Using that k2zh1 − zh0 kL ≤ 2kzh1 kL + kzh0 kL and kzh1 kV ≤ kzh1 + zh0 kV + kzh0 kV , we deduce that there is a constant c such that 1 2 kzh1 k2L + k2zh1 − zh0 k2L + 2∆tαkzh1 k2V ≤ c kzh0 k2L + α∆tkzh0 k2V + ∆t ky k . ′ h V α h ⊓ ⊔ The rest of the proof is similar to that of Lemma 53.2. Theorem 53.14. There is c, independent of h, ∆t, and T , such that, if the solution to (52.4) is in Z(Q) = C 2 ([0, T ]; W ) ∩ C 3 ([0, T ]; V ′ ), ku∆t − uh∆t kℓ∞ ([0,T ];L) ≤ c(hk+1 + ∆t2 )kukZ(Q) , √1 ku∆t − uh∆t kℓ2 ([t1 ,T ];V ) ≤ c 1 + √1 (hk + ∆t2 )kukZ(Q) . T T ⊓ ⊔ Proof. Proceed as in the proof of Theorem ??. 53.3.2 Crank–Nicolson 1 The Crank–Nicolson algorithm is based on the approximation ∂t u(tn+ 2 ) = 1 1 n+1 ) − u(tn )) + O(∆t2 ), where tn+ 2 := tn + ∆t ∆t (u(t 2 . This is a one-step method. The approximate sequence uh∆t ∈ VhN +1 is defined so that the following holds for all vh ∈ Vh and all n ∈ {0:N − 1}: ( 0 uh = Ih u0 , (53.16) 1 1 n+1 1 − unh , vh )L + a(tn+ 2 , 12 (un+1 + unh ), vh ) = (f (tn+ 2 ), vh )L . h ∆t (uh N The stability analysis can be done by defining the operator Sh∆t : N 0 N N N Vh ×Vh −→ Vh so that the sequence (zh , . . . , zh ) = Sh∆t (xh , yh∆t ) is generated according to the rule ( 0 zh = xh , (53.17) 1 n+1 1 − zhn , vh )L + a(tn+ 2 , 12 (zhn+1 + zhn ), vh ) = (yhn+1 , vh )L , ∆t (zh 1 3 1 Defining ρh∆t (f ) := (PhL f (t 2 ), PhL f (t 2 ), . . . , PhL f (tN − 2 )), the algorithm N (53.13) is such that uh∆t = Sh∆t (Ih u0 , ρh∆t (f )). Let us set z̄hn = 12 (zhn +zhn−1). N Lemma 53.15 (Stability). Let zh∆t = Sh∆t (xh , yh∆t ) as in (53.17), then kzhn kL ≤ kzh0 kL + kz̄h∆t kℓ2 ([t1 ,T ];V ) ≤ √1 kyh∆t kℓ2 ([t1 ;tn ];V ′ ) , α h √1 kz 0 kL + 1 kyh∆t kℓ2 ([t1 ,T ];V ′ ) h α α h (53.18) (53.19) 750 Chapter 53. Parabolic PDEs: time approximation Proof. Use ∆t(zni+1 + zhi ) as test function in (53.17) at time ti : kzhi+1 k2L + i+1 ∆tα 4 kzn + zhi k2L2 ≤ kzhi k2L + i+1 2 ∆t α kyh kVh′ . Conclude by summing the above inequality from i = 0 to i = n − 1. ⊓ ⊔ Remark 53.16. Note that, unlike the backward Euler and the BDF2 schemes, the term kzh0 kL on the right-hand side of (53.18) is not multiplied by an exponential decreasing in time. This reflects a slight loss of stability when compared to the other two scheme. The backward Euler, the Crank– Nicolson, and the forward Euler schemes are part of one family of methods parameterized by θ ∈ [0, 1] and consisting of approximating a(t, u, v) by a(tn + θ∆t, (1 − θ)unh + θun+1 , vh ) and ∂t u(t) by (un+1 − unh )/∆t. This soh h called θ-method can be shown to be unconditionally stable when θ ∈ [ 12 , 1] and conditionally stable when θ ∈ [0, 12 ]. The method corresponding at θ = 12 , which is the Crank–Nicolson method, is said to be marginally stable. Note finally that the Crank–Nicolson method is the only one in this family that is second-order accurate in time. ⊓ ⊔ Exercises Exercise 53.1 (Discrete Gronwall). Prove Lemma 53.3. Exercise 53.2. Consider the sequences of non-negative numbers {an }n≥0 , {f n }n≥1 , {g n }n≥1 , and {hn }n≥1 . Assume there is γ ≥ 0 such that √ √ an+1 ≤ an + γf n+1 an + γg n+1 an+1 + γhn+1 , ∀n ∈ N. Prove that there is c, independent of γ, such that, for all N ≥ 0 and p ∈ [1, ∞], 1 1 aN ≤ c a0 + (N γ)2 2 (kf k2ℓp[1,N ] + kgk2ℓp[1,N ] ) + (N γ) 1 khkℓp[1,N ] . Np Np √ (Hint: Fix N , then use f n+1 an ≤ N1γ an + N4γ (f n+1 )2 , ∀n ≤ N − 1.) Exercise 53.3 (Implicit-explicit scheme). Let L, V , and X be three Hilbert spaces such that X ֒→ V ֒→ L ≡ L′ ֒→ V ′ ֒→ X ′ . Let A ∈ L(V ; L) and D ∈ L(X; X ′ ) be two operators. Assume that D is coercive with respect to the X-norm with coercivity constant equal to ℓ2 , i.e., hDu, uiX ′ ,X ≥ ℓ−2 kuk2X for all u ∈ X, and kukL ≤ kukX for all u ∈ X. Let c = ℓ maxu∈V kAukL /kukX . T Let u0 ∈ X and f ∈ L2 ((0, T ); X ′ ). Let N ≥ 1, set ∆t = N , tn = n∆t for n R t 0 ≤ n ≤ N , and hf n , viX ′ ,X := tn−1 hf (t), viX ′ ,X dt for all v ∈ X. Let α > 0 and approximate the problem ∂t u + αDu + Au = f , u(0) = u0 , by using the following scheme: u0 = u0 and (un+1 −un ,v)L +αhDun+1 , viX ′ ,X +(Aun , v)L ∆t = hf n+1 , viX ′ ,X , n ≥ 0, ∀v ∈ X. Part IX. Time-Dependent PDEs 751 (i) Give an example of PDE that can be represented by the above functional framework. (ii) Prove that if 2 cℓ α ≤ 1, then kun+1 k2L +∆tαℓ−2 kun+1 k2X ≤ kun k2L + 21 ∆tαℓ−2 kun k2X + αℓ2−2 ∆tkf n+1 k2X ′ . (iii) Assume now that (Av, v)L ≥ 0 for all v ∈ V and 2∆tc2 ≤ α. Prove that 1 kun+1 k2L +∆tαℓ−2 kun+1 k2X ≤ kun k2L +∆tαℓ−2 kun k2X + ∆tα−1 ℓ2 kf n+1 kX ′ . 2 The desired estimate follows by using the assumption (iv) Derive the corresponding error estimates. (v) Redo questions (i) and (ii) using BDF2, i.e., u0 = u0 and 1 3 n+1 ∆t ( 2 u − un + 12 un−1 ) + αDun+1 + A(2un − un−1 ) = f n+1 , n ≥ 0. (Hint: Use (53.15).) Exercise 53.4 (CFL number). Use the notation of Exercise 53.3 and let (·, ·)X be the inner product in X. Let a be the bilinear form associated with the operator A, a(u, v) := (Au, v)L , a ∈ L(V, L). Let Xh ⊂ X be a finitedimensional subset of X. Let ci (h) = maxvh ∈Xh kvh kX /kvh kL . Approximate the problem ∂t u + Au = f , u(0) = u0 by using the following scheme: u0h ∈ Xh solves (u0h , vh )X = (u0 , vh )X , for all v ∈ Xh , and un+1 ∈ Xh solves h n+1 1 ∆t (uh − unh , vh )L + α(h, ∆t)(unh , vh )X + a(un , vh ) = hf n+1 , vh iX ′ ,X , for all vh ∈ Xh and where α(h, ∆t) ≥ 0 is a so-called artificial viscosity parameter. (i) Explain why the above algorithm can be more attractive than the implicit Euler method obtained with α = 0? 2 2 2 (ii) Prove that if 2 ∆t α (α ci (h) + c ) ≤ 1, then n 2 n+1 2 ∆t kun+1 k2L + α∆tkun+1 k2X ≤ kun k2L + α ∆t kX ′ . 2 ku kX + 2 α kf (iii) Prove that the above stability condition implies 4∆tc ci (h) ≤ 1. The constant ∆tc ci (h) is called the Courant–Friedrichs–Levy (CFL) number. Determine the admissible range for α(h, ∆t). Show that 4∆tc2 is an admissible value for α. (iv) Derive the corresponding error estimates. Exercise 53.5 (Leap-frog). Let V ֒→ L be two Hilbert spaces. Let A ∈ L(V ; L) be a maximal monotone operator, i.e., (Av, v)L ≥ 0 for all v ∈ V (A is monotone), and for all f ∈ L there is v ∈ V such that v + Av = f , (A is R tn+1 1 maximal). Let u0 ∈ V , let f ∈ L2 ((0, T ); L), and set f n = ∆t f (t) dt tn−1 752 Chapter 53. Parabolic PDEs: time approximation for n ≥ 1. Approximate the problem ∂t u + Au = f , u(0) = u0 , by using the so-called leap-frog scheme: ( R ∆t 1 f (t) dt), u0 = u0 , u1 = u0 − ∆t(Au0 − ∆t 0 n+1 1 2∆t (u − un−1 ) + Aun = f n . (i) Prove kun+1 k2L + kun k2L ≤ kun k2L + kun−1 k2L + 4∆tkf n kL (kun k2L + 1 kun−1 k2L ) 2 for all n ≥ 1. (ii) Why is this scheme interesting? (iii) Prove ku∆t kℓ∞ ([0,T ];L) ≤ c(ku0 kL + ∆tkAu0 kL + T kf kL2((0,T );L) ). (iv) Complete the error analysis. Exercise 53.6 (Explicit Euler). Use the notation of Exercise 53.5. Let Vh ⊂ V be a finite-dimensional subspace of V . Let ci (h) = maxvh ∈Vh kAvh kL /kvh kL . T Let N ≥ 1, set ∆t = N and tn = n∆t for 0 ≤ n ≤ N . Let u0 ∈ V , R tn 1 f ∈ L2 ((0, T ); L), and set f n = ∆t f (t) dt. Consider the explicit Euler tn−1 scheme: ( 0 (uh , vh )L = (u0 , vh )L , ∀vh ∈ Vh , n+1 1 ∆t (uh − unh , vh )L + (Aunh , vh )L = (f n+1 , vh )L , ∀vh ∈ Vh . (i) Give a bound on ci (h) when A is a first-order differential operator and Vh is a finite element space based on a quasi-uniform mesh. (ii) Prove that if ∆t ci (h)2 ≤ 1, then kun+1 k2L ≤ kunh k2L + 2∆tkun+1 k2L + ∆tkf n+1 k2L . h h (Hint: Use 2p(p − p) = p2 + (p − q)2 − q 2 and 2pq ≤ γp2 + γ1 q 2 for all γ > 0.) (iii) Assuming ∆t ci (h)2 ≤ 1, prove the estimate kuh∆t kℓ∞ ([0,T ];L) ≤ c(T )(kf kL2 ((0,T );L) + ku0 kL ).