Parabolic PDEs: time approximation Implicit Euler

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1 Part IX, Capter 53 Parabolic PDEs: time approximation We are concerned in tis capter wit bot te time and te space approximation of te model problem (52.4). We adopt te metod of line introduced in and approximate te semi-discrete problem (52.3) wit respect to time. In te entire capter we keep te same notation and conventions as in Capter Implicit Euler We introduce te backward Euler metod to approximate te semi-discrete problem (52.3) and investigate te stability and convergence properties of te resulting algoritm. For tis section and te rest of te capter we set = N were N is a nonzero natural number Principle Let u 0 V be a reasonable approximation of u 0. We formalize tis constructionbyassumingtattereisanoperatori L(L;V )suctatu 0 := I u 0. We moreover asume tat (I ) >0 is uniformly bounded in L(L;L). For instance we can take I = P L, were P L is te L-ortogonal projection from L to V, i.e., (P L (v),v ) L = (v,v ) L for all v V and all v L. We will abuse te notation by denoting again P L te extension of P L to V, i.e., (P L f,v ) L = f,v V,V. Note tat tis make sense since V V and V is finite-dimensional, i.e., sup 0 v V v V / v L <. e implicit Euler (or backward Euler metod) consists of constructing an approximating sequence u = (u 0,...,un,...uN ), un V, n {0:N}, by mean of te following time-stepping algoritm: v V, (u n+ u n,v ) L +a(t n+,u n+,v ) = (P L f(t n+ ),v ) L, (53.)

2 742 Capter 53. Parabolic PDEs: time approximation were we expect u n+ to be a reasonable approximation of u(t n+ ), were t n+ = t n + = (n+), 0 n N. o analyze te properties of tis algoritm, we introduce te operator S N : V V N N+ V suctat, forx V andy V N,te sequence z = (z 0,...,zN ) = SN (x,y ) is inductively defined by z 0 = x and v V, (z n+ z n,v ) L +a(t n+,z n+,v ) = (y n+,v ) L. (53.2) Upon setting ρ (f) = (P L f(t ),...,P L f(t N )), we realize tat (53.) can be re-written as follows: u = S N (I (u 0 ),ρ (f)). Note tat we abuse te notation and indifferently write z = (z 0,...,zN ) or z = (z,...,zn ) wen te context is unambiguous. Remark 53.. Note tat we assumed tat f C 0 ((0,];V ) for (53.) to make sense. is assumption is made in te entire capter. It can be lifted in te context of te backward Euler metod by replacing f(t n+ ) by t n+ t n f(τ)dτ Stability We define y V := sup 0 z V (y,z ) L / z V. Let n 0,n {0:N} and let B be a normed space; we additionally consider te following discrete norms: y l 2 ([t n 0,t n ];B) := ( n=n n=n 0 y n 2 B ) 2, (53.3) y l ([t n 0,t n ];B) := max n 0 n n y n B. (53.4) e operator S N as stability properties tat are similar to tose of te continuous problem stated in eorem 52.2 (i.e., continuous dependence wit respect to te data). Lemma 53.2 (Stability). Assume and let z αc 2 = S N (x,y ) P be defined in (53.2). en, te following olds for any n {0:N}: z n L z l 2 ([t,t N ];V) { e 4 αc2 P tn x L + α y l 2 ([t,t n ];V ), e 4 αc2 P tn x L + y l ([t,t n ];V ), { α x L + α y l 2 ([t,];v ), α x L + α y l ([t,];v ). (53.5) (53.6)

3 Part IX. ime-dependent PDEs 743 Proof. () We test (53.2) wit 2z n+. e relation 2p(p q) = p 2 +(p q) 2 q 2 and te coercivity of a imply tat z n+ 2 L + z n+ z n 2 L +2α z n+ 2 V z n 2 L +2 y n+ V z n+ V z n 2 L + +α z n+ 2 V. α yn+ 2 V e last inequality is obtained from te aritmetic geometric inequality (A.4). e inequality c P z n+ L z n+ V leads to (+αc 2 P) z n+ 2 L z n+ 2 L+α z n+ 2 V z n 2 L+ α yn+ 2 V. (53.7) e Discrete Gronwall Lemma (see Lemma 53.3 below), in turn, implies tat z n+ 2 L x 2 L (+αc 2 P )n+ + α n (+αc 2 k=0 P )n k+ yk+ 2 V. (53.8) Note tat ( + αc 2 P ) e 2 αc2 P if αc 2 P. Moreover, ( + αc 2 P ) <. e first estimate in (53.5) follows easily. e second estimate follows from (53.8) by using te inequality n k=0 (+αc 2 P )n k+ αc 2. P (2) Summing (53.7) from n = 0 to N leads to α N z n 2 V x 2 L + N α y n 2 V, n= n= wence te estimates in (53.6) are readily deduced. Lemma 53.3 (Discrete Gronwall). Let γ and β be two real numbers. Let (a n ) n 0, (f n ) n 0 be two sequences suctat (+γ)a n+ a n +βf n. en, a n+ a 0 n (+γ) n+ +β f k (+γ) n k+. Remark (i) Since u = S N (I (u 0 ),(P L f) ), Lemma 53.2, togeter wit te bound c P f V f L, implies tat te solution to (53.) satisfies te following stability estimates: k=0 u n L e 4 αc2 P tn u 0 L + αc 2 P f C 0 ([0,t n ];L), u l 2 ([t,];v) α u 0 L + f C 0 ([0,];L). (ii) e assumption is just made to make te estimates (53.5)- αc 2 P (53.6)looknice; it is not astabilityrestrictionon te time step. e algoritm is said to unconditionally stable.

4 744 Capter 53. Parabolic PDEs: time approximation Error analysis eorem Letube te solution to (52.4). Assumingtat u C 2 [0,];V ) C ([0,];L) we ave u(t n ) u n L e 4 αc2 P tn e 0 L + η C 0 ([0,t n ];L) + η αc 2 C ([0,t n ];L) + P u C 2 ([0,t n ];V ) e l 2 ([t,t N ];V) α e 0 L + η C 0 ([0,];L) + η C ([0,];L) + α u C 2 ([0,];V ). Proof. We are going to proceed like in te proof of eorem () Let P t be te elliptic projector defined in (52.5), let w (t) := P t (u) for any t [0,], and denote w n := w (t n ) for any n {0:N}. By definition, w 0 = P 0(u). e equation (52.3) can be re-written as follows: (w n+ w n,v ) L +a(t n+,w n+,v ) = (f(t n+ ),v ) L +(P L R n+,v ) L, (53.9) t n+ t n+ were R n+ := (d t n τ w (τ) d τ u(τ))dτ (τ t n )d t n ττ u(τ)dτ. Let us now introduce e n := wn un and ηn = u(tn ) w n. Subtracting (53.) from (53.9), we obtain tat (e n+ e n,v ) L +a(t n+,e n+,v ) = (P L R n+,v ) L, v V. Setting R = (P L R,...,P L R N ), te above argument sows tat e = S N (e 0,R ). We conclude by invoking te stability Lemma 53.2: e n L e 4 αc2 P tn e 0 L + R l ([0,t n ];V ), e l 2 ([t,t N ];V) α e 0 L + α R l ([0,];V ). We (rougly) estimate R n+ V as follows: R n+ V c P η C ([t n,t n+ ];L) + u C2 ([t, t n+ ];V ), wic immediately implies tat u(t n ) u n L e 4 αc2 P tn e 0 L + η C 0 ([0,t n ];L) + η αc 2 C ([0,t n ];L) + P u C2 ([0,t n ];V ) e l 2 ([t,t N ];V) α e 0 L + η C 0 ([0,];L) e conclusion follows readily. + η C ([0,];L) + α u C 2 ([0,];V ).

5 Part IX. ime-dependent PDEs 745 Example 53.6 (Finite elements). Let us assume tat V is composed of conforming finite elements of degree k, see Let u be solution of te eat equation (or tat of te equation wit te bilinear form (52.2)) and asume tat u C ([0,];H k+ (D)) C 2 ([0,];L 2 (D)). Assume also tat te adjoint of te differential operator associated wit te elliptic projector as a smooting property in H +s (D), s [0,], (see e.g., eorem??), ten tere is c, uniform wit respect to and, suc tat u P t (u) C ([0,];L 2 (D)) c k+s u C ([0,];H k+ (D)) u P t (u) C 0 ([0,];H (D)) c k u C 0 ([0,];H k+ (D)). Assumingtatu 0 iswellapproximated,say e 0 L 2 ck+s,tereiscuniform wit respect to, u and suc tat te following error estimates old: u u l ([0,];L (D)) c( k+s u C ([0,];H k+ (D)) + u C2 ([0,];L 2 (D))) u u l 2 ([t,];h (D)) c( k u C ([0,];H k+ (D)) + u C 2 ([0,];L 2 (D))). Remark 53.7 (Optimality). e above error analysis is by no mean optimal. Our goal in tis capter is just to sow te key mecanisms in play. Remark 53.8 ( + ). Note tat te constant c is independent of. Hence, if u C ([0,];H k+ (D)) and u C 2 ([0,];L 2 (D)) are bounded wit respect to, te error estimates are uniform in, i.e., te error is uniformly controlled for arbitrary large times. is remarkable property is caracteristic of parabolic equations: tese equations loose memory of te initial data, and tey also progressively loose memory of approximation errors made in te past, i.e., te approximation error does not accumulate as time grows Principle 53.2 Explicit Euler We start as in te backward Euler metod by defining u 0 := I u 0 V to be a reasonable approximation of u 0. e forward Euler (or explicit Euler ) algoritm consists of approximating te solution to (52.4) by constructing a sequence u = (u 0,...,uN ), un V, n {0:N}, suc tat u n+, 0 n N, solves te following problem: v V, (u n+ u n,v ) L +a(t n+,u n,v ) = (f(t n+ ),v ) L. (53.0) o investigate te stability of te metod, we now define te operator S N : V V n N+ V suc tat te sequence z = (z 0,...,zN ) :=

6 746 Capter 53. Parabolic PDEs: time approximation S N (x,y ) is given by z 0 = x and solving te following sequence of problems for all n {0:N }: v V, (zn+ z n,v ) L +a(t n+,z n,v ) = (y n+,v ) L. (53.) isdefinitionimpliestat,uponsettingρ (f) := (P L f(t ),...,P L f(t N )), te sequence solving (53.0) satisfies u = S N (I (u 0 ),ρ (f)) stability e analysis below will reveal tat stability of te algoritm is controlled by te following parameter: c i () = max 0 v V v V v L. (53.2) is quantity is finite since V is finite-dimensional. If V is a finite element space based on a quasi-uniform mes sequence, an inverse inequality sows tat c i () c, see Lemma??. Recall tat tere is M suc tat a(t,v,w) M v V w V for all t [0,], and all v,w V. Lemma 53.9 (Stability). Let θ (0,) be a real number. Let z = S N (x θα,y ) be defined in (53.). Assume tat c i() 2 M, ten 2 z n L x L e 4 αc2 P ( θ)tn + θ y n l ([0,t n ];V ), z l 2 ([t,];v) x L + α( θ) α ( θ) yn l ([0,];V ). Proof. We proceed as in te proof of Lemma esting (53.) wit 2z n+ and using te relation 2p(p q) = p 2 +(p q) 2 q 2 togeter wit (A.4), we obtain z n+ 2 L + zn+ z n 2 L +2a(tn+,z n,zn+ ) z n 2 L +2 yn+ V z n+ V z n 2 L + Furtermore, te inverse inequality (53.2) implies α yn+ 2 V +α z n+ 2 V. a(t n+,z,z n n+ ) = a(t n+,z n+,z n+ )+a(t n+,z n z n+,z n+ ) α z n+ 2 V M zn zn+ V z n+ V α z n+ 2 V c i()m z n+ z n L z n+ V As a result, z n+ 2 L + ( α z n+ 2 V M2 c i() 2 2θα z n+ z n 2 L θα 2 zn+ 2 V. M2 c i() 2 θα ) z n+ z n 2 L +α( θ) zn+ 2 V z n 2 L + α yn+ 2 V.

7 Part IX. ime-dependent PDEs 747 Since θα M 2 c i() 2, we infer tat z n+ 2 L +α( θ) z n+ 2 V z n 2 L + α yn+ 2 V. e rest of te proof is te same as tat of Lemma Remark 53.0 (Conditional stability). e forward Euler algoritm is said to be conditionally stable because of te restriction on Error analysis eorem 53.. Let u be te solution to (52.4). Assumetat u C 2 ([0,];V ) C α [0,];V) and < c i() 2 M, ten 2 u(t n ) u n L e 0 L e 4 αc2 P ( θ)tn + θ c(t n,u), u u l 2 ([t,];v) e 0 α( θ) L + c(,u), α ( θ) were c(t,u) = c P η C ([0,t];L) + u C 2 ([0,t];V ) +M u C [0,t];V) Proof. We proceed as in te proof of eorem First, we investigate te consistency of te metod. Let w (t) := P t (u) were P t is te elliptic projector defined in (52.5) and denote w n := w (t n ) for all n {0:N}. e definition of w n+ implies tat (52.3) can be re-written as follows: (wn+ w n,v ) L +a(t n+,w n,v ) = (f n+,v ) L +(R n+,v ) L, v V, were, R n+ V, 0 n N, is defined so tat te following olds (R n+,v ) L = ( (wn+ w n ) tu(t n+ ),v )L a(tn+,u(t n+ ) u(t n ),v ), for all v V. is definition implies tat e = S N (e0,r ). e quantity R n+ V is (rougly) estimated by R n+ V c P η C ([t n,t n+ ];L) + u C 2 ([t, t n+ ];V ) +M u C [t n,t n+ ];V). e rest of te proof is te same as tat of eorem Example 53.2 (Finite elements). Using te same assumptions as in Example 53.6, we infer tat tere is c uniform wit respect to, u and suc tat te following error estimates old if < α c i() 2 M 2 : u u l ([0,];L 2 (D)) c( k+s u C ([0,];H k+ (D)) + u C 2 ([0,];L 2 (D))) u u l 2 ([t,];h (D)) c( k u C ([0,];H k+ (D)) + u C 2 ([0,];L 2 (D))).

8 748 Capter 53. Parabolic PDEs: time approximation 53.3 Second-order implicit scemes We present two scemes wit second-order accuracy in time in tis section BDF2 e so-called second-order backward Euler metod is based on te approximation t u(t n+ ) = 2 (3u(tn+ ) 4u(t n ) u(t n )) + O( 2 ). is is a two-step metod, usually referred to as BDF2 in te literature, were te approximate sequence u V N+ is defined so tat te following olds for all v V and all n {0:N }: u 0 = I u 0 (u u 0,v ) L +a( 2, 2 (u +u 0 ),v ) = (P L f(t 2 ),v ) L, (53.3) 2 (3un+ 4un +u n,v ) L +a(t n+,u n+,v ) = (f(t n+ ),v ) L. e stability analysis can be done by defining te operator S N : V V N V N so tat te sequence (z0,...,zn ) = SN (x,y ) is generated according to te rule z 0 = x, (z z,v 0 ) L +a( 2, 2 (z +z),v 0 ) = (y,v ) L, (53.4) 2 (3zn+ 4zn +z n,v ) L +a(t n+,z n+,v ) = (y n+,v ) L, Definingρ (f) := (P L f(t 2),P L f(t 2 ),...,P L f(t N )),tealgoritm(53.3) is suc tat u = S N (I u 0,ρ (f)). Lemma 53.3 (Stability). Let z = S N (x,y ) as in (53.4). en, z n L c( x V + y L )e 4 αc2 P tn + y l ([t,t n ];V ), z l 2 ([t,];v) c α ( x V + y L )+ α y l ([t,];v ). Proof. est te tird equation in (53.4) wit 4z n+. Using te relation 2(a n+,3a n+ 4a n +a n ) = a n a n+ a n 2 + δ tt a n+ 2 wit δ tt a n+ := a n+ 2a n +a n, yields a n 2 2a n a n 2, (53.5) z n+ 2 L + 2zn+ z n 2 L + δ ttz n+ 2 L +4α zn+ 2 V zn 2 L 2 L +2α zn+ 2 V + 2 us, for n, + 2z n zn α yn+ 2 V.

9 Part IX. ime-dependent PDEs 749 z n+ 2 L + 2zn+ z n 2 L +2α zn+ 2 V zn 2 L + 2z n zn 2 L + 2 α yn+ 2 V. Furtermore, testing te second equation in (53.4) wit (z +z0 ) leads to z 2 L + α 4 z +z0 2 V z0 2 L + α y 2 V. Using tat 2z z0 L 2 z L+ z 0 L and z V z +z0 V + z 0 V, we deduce tat tere is a constant c suc tat ) z 2 L + 2z z0 2 L +2α z 2 V ( z c 0 2 L +α z0 2 V + α y 2 V. e rest of te proof is similar to tat of Lemma eorem ere is c, independent of,, and, suc tat, if te solution to (52.4) is in Z(Q) = C 2 ([0,];W) C 3 ([0,];V ), u u l ([0,];L) c( k+ + 2 ) u Z(Q), ( u u l 2 ([t,];v) c + )( k + 2 ) u Z(Q). Proof. Proceed as in te proof of eorem?? Crank Nicolson e Crank Nicolson algoritm is based on te approximation t u(t n+ 2) = (u(tn+ ) u(t n )) + O( 2 ), were t n+ 2 := t n + 2. is is a one-step metod. e approximate sequence u V N+ is defined so tat te following olds for all v V and all n {0:N }: { u 0 = I u 0, (un+ u n,v ) L +a(t n+ 2, 2 (un+ +u n ),v (53.6) ) = (f(t n+ 2 ),v ) L. e stability analysis can be done by defining te operator S N : V V N V N so tat te sequence (z0,...,zn ) = SN (x,y ) is generated according to te rule { z 0 = x, (zn+ z n,v ) L +a(t n+ 2, 2 (zn+ +z n ),v (53.7) ) = (y n+,v ) L, Defining ρ (f) := (P L f(t 2 ),PL f(t 3 2 ),...,PL f(t N 2 )), te algoritm (53.3)issuctatu = S N (I u 0,ρ (f)). Letusset z n = 2 (zn +zn ). Lemma 53.5 (Stability). Let z = S N (x,y ) as in (53.7), ten z n L z 0 L + α y l2 ([t ;t n ];V ), (53.8) z l2 ([t,];v) α z 0 L + α y l2 ([t,];v ) (53.9)

10 750 Capter 53. Parabolic PDEs: time approximation Proof. Use (z i+ n +z i ) as test function in (53.7) at time ti : z i+ 2 L + α 4 zi+ n +z i 2 L 2 zi 2 L + α yi+ 2 V. Conclude by summing te above inequality from i = 0 to i = n. Remark Note tat, unlike te backward Euler and te BDF2 scemes, te term z 0 L on te rigt-and side of (53.8) is not multiplied by an exponential decreasing in time. is reflects a sligt loss of stability wen compared to te oter two sceme. e backward Euler, te Crank Nicolson, and te forward Euler scemes are part of one family of metods parameterized by θ [0,] and consisting of approximating a(t,u,v) by a(t n + θ,( θ)u n + θun+,v ) and t u(t) by (u n+ u n )/. is socalled θ-metod can be sown to be unconditionally stable wen θ [ 2,] and conditionally stable wen θ [0, 2 ]. e metod corresponding at θ = 2, wic is te Crank Nicolson metod, is said to be marginally stable. Note finally tat te Crank Nicolson metod is te only one in tis family tat is second-order accurate in time. Exercises Exercise 53. (Discrete Gronwall). Prove Lemma Exercise Consider te sequences of non-negative numbers {a n } n 0, {f n } n, {g n } n, and { n } n. Assume tere is γ 0 suc tat a n+ a n +γf n+ a n +γg n+ a n+ +γ n+, n N. Provetat tereis c, independent ofγ,suc tat, forall N 0and p [, ], ( a N c a 0 +(Nγ) 2 ( f 2 l p [,N] + g 2 l p [,N] )+(Nγ) ). N 2 p N l p [,N] p (Hint: Fix N, ten use f n+ a n Nγ an + Nγ 4 (fn+ ) 2, n N.) Exercise 53.3 (Implicit-explicit sceme). Let L, V, and X be tree HilbertspacessuctatX V L L V X.LetA L(V;L)and D L(X;X ) be twooperators.assume tat D is coercivewit respectto te X-norm wit coercivity constant equal to l 2, i.e., Du,u X,X l 2 u 2 X for all u X, and u L u X for all u X. Let c = lmax u V Au L / u X. Let u 0 X and f L 2 ((0,);X ). Let N, set = N, tn = n for 0 n N, and f n,v X,X := t n t f(t),v n X,X dt for all v X. Let α > 0 and approximate te problem t u+αdu+au = f, u(0) = u 0, by using te following sceme: u 0 = u 0 and (u n+ u n,v) L +α Du n+,v X,X+(Au n,v) L = f n+,v X,X, n 0, v X.

11 Part IX. ime-dependent PDEs 75 (i) Give an example of PDE tat can be represented by te above functional framework. (ii) Prove tat if 2 cl α, ten u n+ 2 L +αl 2 u n+ 2 X un 2 L + 2 αl 2 u n 2 X + 2 αl 2 f n+ 2 X. (iii) Assume now tat (Av,v) L 0 for all v V and 2c 2 α. Prove tat u n+ 2 L+αl 2 u n+ 2 X u n 2 L+αl 2 u n 2 X+ 2 α l 2 f n+ X. e desired estimate follows by using te assumption (iv) Derive te corresponding error estimates. (v) Redo questions (i) and (ii) using BDF2, i.e., u 0 = u 0 and (3 2 un+ u n + 2 un )+αdu n+ +A(2u n u n ) = f n+, n 0. (Hint: Use (53.5).) Exercise 53.4 (CFL number). Use te notation of Exercise 53.3 and let (, ) X be te inner product in X. Let a be te bilinear form associated wit te operator A, a(u,v) := (Au,v) L, a L(V,L). Let X X be a finitedimensional subset of X. Let c i () = max v X v X / v L. Approximate te problem t u+au = f, u(0) = u 0 by using te following sceme: u 0 X solves (u 0,v ) X = (u 0,v ) X, for all v X, and u n+ X solves (un+ u n,v ) L +α(,)(u n,v ) X +a(u n,v ) = f n+,v X,X, for all v X and were α(,) 0 is a so-called artificial viscosity parameter. (i) Explain wy te above algoritm can be more attractive tan te implicit Euler metod obtained wit α = 0? (ii) Prove tat if 2 α (α2 c i () 2 +c 2 ), ten u n+ 2 L +α u n+ 2 X u n 2 L +α 2 un 2 X +2 α fn+ 2 X. (iii) Prove tat te above stability condition implies 4cc i (). e constant cc i () is called te Courant Friedrics Levy (CFL) number. Determine te admissible range for α(,). Sow tat 4c 2 is an admissible value for α. (iv) Derive te 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 tere is v V suc tat v + Av = f, (A is maximal). Let u 0 V, let f L 2 ((0,);L), and set f n = t n+ f(t)dt t n

12 752 Capter 53. Parabolic PDEs: time approximation for n. Approximate te problem t u+au = f, u(0) = u 0, by using te so-called leap-frog sceme: { u 0 = u 0, u = u 0 (Au 0 f(t)dt), 0 2 (un+ u n )+Au n = f n. (i) Prove u n+ 2 L + un 2 L un 2 L + un 2 L + 4 fn L ( u n 2 L + u n 2 L ) 2 for all n. (ii) Wy is tis sceme interesting? (iii) Prove u l ([0,];L) c( u 0 L + Au 0 L + f L 2 ((0,);L)). (iv) Complete te error analysis. Exercise 53.6 (Explicit Euler). UsetenotationofExercise53.5.LetV V beafinite-dimensionalsubspaceofv.letc i () = max v V Av L / v L. Let N, set = N and tn = n for 0 n N. Let u 0 V, t n f L 2 ((0,);L), and set f n = f(t)dt. Consider te explicit Euler t n sceme: { (u 0,v ) L = (u 0,v ) L, v V, (un+ u n,v ) L +(Au n,v ) L = (f n+,v ) L, v V. (i) Give a bound on c i () wen A is a first-order differential operator and V is a finite element space based on a quasi-uniform mes. (ii) Prove tat if c i () 2, ten u n+ 2 L u n 2 L +2 u n+ 2 L + f n+ 2 L. (Hint: Use 2p(p p) = p 2 + (p q) 2 q 2 and 2pq γp 2 + γ q2 for all γ > 0.) (iii) Assuming c i () 2, prove te estimate u l ([0,];L) c()( f L 2 ((0,);L) + u 0 L ).