Convergence analysis for f = f(x) - The Strang Splitting Method for Semilinear Parabolic Proble

4.4 Convergence analysis for a solution independent source term

In this section, we give an estimate of the error of the splitting (4.6) when the source term f =f(x) only depends on the space variable x. We assume f 2L^p(⌦), where again 1< p <1. We can write the parabolic problem (4.1) as

@tu(x, t) = Du(x, t) +f(x) in ⌦⇥(0,1), Bu(x, t) = b(x) on @⌦⇥(0,1),

u(x,0) = u0(x) in⌦, (4.17)

Since we are interested in the semi-discretization in time, for brevity of notations, we write, from now on, u(t) andf instead ofu(x, t) andf(x). We denote byr(y) the stability function of the Crank-Nicolson method given by,

r(y) = 1 + ^y₂

1 ^y₂. (4.18)

We recall that, for any Runge-Kutta method applied with a time step ⌧ > 0 to the Dahlquist scalar test equation (see [35, Definition IV.2.1])

dt(t) = x(t), x(0) =x₀,

with 2 C, one obtains the induction xn+1 = R(h )xn, where R : C ! C is a rational approximation of the exponential that we call the stability function of the Runge-Kutta method. For a fixed degree of the numerator and denominator, the rational approxima-tions that have the highest order of approximation are called the Padé approximaapproxima-tions of the exponential (cf. [35, Chapter IV.3]) and efficient Runge-Kutta methods are typically constructed to have a stability function equal to a Padé approximation. The approxi-mation r(z) that corresponds to the Crank-Nicolson stability function is the (1,1)-Padé approximation (4.18). An A-stable method is by definition a Runge-Kutta method whose stability function verifies |R(y)|1 for all y 2C ={z 2 C ; <(z) 0}. We recall that the only Padé approximations that verify this property are the (j, k)-Padé approximations with k j k+ 2 (see [35, Theorem 4.12]).

Remark 4.4.1. Consider the following ordinary di↵erential equation dx

dt(t) = x(t) +b, x(0) =x0,

with , b 2C. A Runge-Kutta method with stability function R(y) yields xn+1 =R(h )xn+R(h ) 1

b. (4.19)

The Strang splitting x_n+1 = ^bh 2

,RK

h b

h 2

(x_n), where ^bh 2

(x) = x+ ^h₂b and _h^,RK(x) = R(h )x yields

xn+1 =R(h )(xn+ h 2b) + h

2b. (4.20)

56 CHAPTER 4. STRANG SPLITTING WITH CRANK-NICOLSON Equations (4.19) and (4.20) coincide if and only if

R(y) = r(y) = 1 + ^y₂ 1 ^y₂,

which is the stability function (4.18) of the Crank-Nicolson scheme. Therefore, the only Runge-Kutta methods for which equations (4.19) and (4.20) coincide are the ones with a stability function equal to r(y).

Remark 4.4.2. One step of the Crank-Nicolson scheme applied to the parabolic prob-lem (4.1) is given by the solution un+1 of the following equation,

un+1 un

⌧ =Dun+1+un

2 +f(un+1) +f(un)

2 in ⌦,

Bun+1+un

2 =b on @⌦. (4.21)

We observe that, for f = f(x), formula (4.21) is equal to formula (4.16) which describes one step of the splitting (4.6). Therefore, we deduce that for f =f(x), the Strang splitting with Crank-Nicolson (4.6) is equivalent to the Crank-Nicolson scheme itself applied to the whole problem (4.17).

4.4.1 Main results

The main result of this chapter is the following theorem, which states that the splitting (4.6) yields a method of order 2 of accuracy away from a neighbourhood of the origin t = 0 on unbounded intervals (t >0). In contrast, in a neighbourhood of zero, the order of accuracy reduces to one.

Theorem 4.4.3. Let e_n = u_n u(t_n) be the global error of the splitting method (4.6) applied to the parabolic problem (4.17), where u0 2W^2,p(⌦) and satisfies Bu0 =b on @⌦.

Then, for all n 0, the global error en is given by

en = r(⌧A)ⁿ e^n⌧A A ¹(Du0+f) (4.22) and it satisfies the bound

kenk^L^p^(⌦)  C⌧² t_n , where C is a constant independent of ⌧, n, and tn=n⌧.

Remark 4.4.4. The estimate of Theorem 4.4.3 could be used to derive a fully discrete estimate of the form

ku^h_n u(tn)k C(⌧²/tn+h^p)

where u^h_n denotes a standard finite element discretization of order pon a spatial mesh with size h. The idea of the proof is to rely on the triangle inequality

ku^h_n u(tn)k  ku^h_n u^h(tn)k+ku^h(tn) u(tn)k

where u^h(t_n) denotes the semi-discretization in space at time t_n. Then, observe that the estimate of Theorem 4.4.3 also holds for u^h_n u^h(tn) uniformly with respect to the spatial mesh size h, using that the space discretization of the di↵usion operator A is a self-adjoint operator that satisfies assumptions analogous to (4.9) and (4.10), see e.g. [20].

4.4. CONVERGENCE ANALYSIS FOR f =f(x) 57 Remark 4.4.5. One can extend the framework described in Section 2 to more abstract problems and consider for example, a problem where D is the bi-Laplacian with appropri-ate boundary conditions. This formulation also include Galerkin approximation of parabolic problems (see [58]). More generally, for a complex separable Banach space X, we need A:D(A)!X, to be a closed densely defined linear operator satisfying the conditions (4.9) and (4.10) or equivalently we require A to be the infinitesimal generator of a uniformly bounded analytic semigroup ( [54, Theorems 2.5.2]). We require f : X !X to be contin-uously di↵erentiable. The operator D : D(D) ! X is assumed to be an extension of A, i.e. D(A) ⇢ D(D). We require z 2 D(D) and u0 z 2 D(A). Then, the problem (4.11) has a unique solution u˜2C¹([0, T], X)\C([0, T],D(A)) ifT is sufficiently small (see [50, Proposition 7.1.10]). The main problem (4.1), for u= ˜u+z becomes

@tu(t) =Du(t) +f(u(t)), for t2(0, T], u(t) z 2D(A) for t2(0, T], u(0) =u0. For this problem, the splitting (4.6), is defined as above for (4.1), where the boundary conditions of the problem (4.1) have been replaced by u(t) z 2 D(A). Under all those conditions, for f =f(x), the convergence analysis (Theorem 4.4.3) remains true, i.e. the splitting method (4.6) has no reduction of order away from the origin and the global error is given by formula (4.22).

Remark 4.4.6. Since the splitting method (4.6) applied to the parabolic problem (4.17) is equivalent to the Crank-Nicolson scheme, it must preserve exactly the stationary states.

Precisely, for Du0 +f = 0, the error satisfies en = 0. For general nonlinearities that depend on the solution f = f(u), the stationary states are not preserved. This is not surprising since in [51], it is proved that a method that is not a Butcher-series method, like the splitting methods, and which is invariant by an affine change of variable (precisely affine-equivalent), cannot preserve all stationary states.

4.4.2 Preliminaries

We give some basic properties of the rational approximationr(y) in (4.18). The following properties are obtained with straightforward computations.

Lemma 4.4.7. We have the following formulas, r(y) + 1 = 2

1 ^y₂, r(y) 1 = y

1 ^y₂, (4.23)

and

(r(y) e^y) = y

2(r(y) + 1) (e^y 1). (4.24)

The rational approximation r(y) satisfies the following result, in the context of homo-geneous parabolic problems, which is proved in the Appendix 4.6 (see [37, Theorem 2.1]

for a proof in a more general case).

Theorem 4.4.8. For u0 2 D(A), where A satisfies the assumptions of Section 4.2 and r(y) is defined in (4.18), we have the following error estimate,

k(r(⌧A)ⁿ e^⌧nA)u₀kL^p(⌦)  C⌧²

tn kAu₀kL^p(⌦),

58 CHAPTER 4. STRANG SPLITTING WITH CRANK-NICOLSON where C is a constant independent of u0, ⌧, n and tn =n⌧.

Note that this corresponds to an estimate of the splitting (4.6) for the specific case where the problem is homogeneous, i.e. f = 0, and with homogeneous boundary condi-tions. In what follows, we give an exact representation of the numerical solution of the splitting (4.6) in term of the operator A. We recover homogeneous boundary conditions with an appropriate change of variable. Let z 2 W^2,p(⌦) be the same function chosen in (4.12) and satisfyingBz =b on@⌦. Defining ˜v1 =v1 z, we rewrite equation (4.14) as follows,

2˜v1 u0

⌧ + 2

⌧z =A˜v1+Dz,

where we recall that the homogeneous boundary conditions are included in the domain of A. This gives the following expression for ˜v1,

We denote u₁ = ^D,CN_⌧ (u₀), the solution of the problem (4.13). Therefore, from equa-tion (4.15), we have the following formula,

u₁ = ^D,CN_⌧ (u₀) = 2˜v₁ u₀+ 2z

Using the equalities (4.23), we obtain

D,CN

Using the following representation of the exact flow ^f_t forf =f(x),

t(u0) = u0+tf, (4.26)

and formula (4.25) for ^D,CN_⌧ , we obtain the following expression for S⌧(un):

S^⌧(un) =z+r(⌧A)(un z) + (r(⌧A) I)A ¹Dz+ ⌧

2(r(⌧A) +I)f. (4.27) Remark 4.4.9. Take any A-stable Runge-Kutta method and denote by R(z) its stability function. Then if this Runge-Kutta method is used to solve the di↵usion equation (4.3), instead of the Crank-Nicolson method, one obtains formula(4.27)for the numerical solution withr(⌧A)replaced withR(⌧A). Indeed, letube the solution of the di↵usion equation (4.3).

Let z 2W^2,p(⌦) be a function satisfying Bz =b on @⌦. Then, we have @tu =A(u z + A ¹Dz)fort >0. Defining y=u z+A ¹Dz, we obtain the following equivalent problem,

@ty=Ay for t >0, y(0) =u0 z+A ¹Dz.

Applying one step of the Runge-Kutta method with initial condition y(0) and with a time step ⌧ gives,

y₁ =R(⌧A)(u₀ z+A ¹Dz).

Hence, we obtain the same formula (4.25)withr(⌧A)replaced withR(⌧A)for the numerical solution of the di↵usion (4.3),

u₁ =R(⌧A)(u₀ z+A ¹Dz) +z A ¹Dz =z+R(⌧A)(u₀ z) + (R(⌧A) 1)A ¹Dz.

4.4. CONVERGENCE ANALYSIS FOR f =f(x) 59

4.4.3 Local error

We start with the following proposition that gives an exact representation of the local error.

Proposition 4.4.10. The local error n+1 = S⌧(u(tn)) u(tn+1) of the splitting (4.6) satisfies the following identity,

n+1 = (r(⌧A) e^⌧A)A ¹e^tⁿ^A(Du0+f).

Proof. We have the following representation of n+1:

n+1 = (r(⌧A) e^⌧A)(u(t_n) z) + (r(⌧A) e^⌧A)A ¹Dz +⌧

2(r(⌧A) +I)f A ¹(e^⌧A I)f. (4.28)

From formula (4.24), we obtain,

n+1 = (r(⌧A) e^⌧A)(u(tn) z+A ¹Dz+A ¹f). (4.29) From equation (4.11), we know that ˜u(tn) is the solution of the following problem,

@_tu(t) =˜ A˜u(t) +f+Dz for t2(0, T], u(0) =˜ u₀ z.

Hence, n+1 satisfies

n+1 = (r(⌧A) e^⌧A)A ¹@tu(t˜ n).

Using the variation of constant formula, we obtain,

@tu(t˜ n) =Au(t˜ n) +f+Dz

=Ae^tⁿ^Au˜0+A Z tn

e^(tⁿ ^s)A(f +Dz)ds +f+Dz

=Ae^tⁿ^Au˜₀+ (e^tⁿ^A I)(f +Dz) +f +Dz

= e^tⁿ^A(A˜u₀+f +Dz)

= e^tⁿ^A(Du₀+f)

where we use A˜u0+Dz =D(˜u0+z) =Du0. This concludes the proof.

Remark 4.4.11. Assume that another Runge-Kutta method, with stability function R(y), is used to solve the di↵usion equation (4.3) involved in the splitting method (4.6). Then, from Remark 4.4.9, we observe that the local error n+1 still satisfies formula (4.28) with r(⌧A) replaced with R(⌧A). However, the representation (4.29) is specific to Runge-Kutta methods having r(y) as a stability function. Indeed, to find this representation of the local error, we used formula (4.24) which is a property satisfied only by the (1,1)-Pad´e approximation (4.18).

60 CHAPTER 4. STRANG SPLITTING WITH CRANK-NICOLSON

4.4.4 Global error

We are now in position to prove Theorem 4.4.3 for the global erroren =un u(tn) of the splitting (4.6). We observe that

en+1 =S^⌧(un) S^⌧u(tn) +S^⌧u(tn) u(tn+1) =S^⌧(un) S^⌧u(tn) + n+1. Proof of Theorem 4.4.3. From formula (4.27) for S⌧(un), we obtain

S^⌧(un) S^⌧u(tn) =r(⌧A)(un u(tn)).

Therefore, we deduce

en+1 =r(⌧A)en+ n+1.

Hence, since e0 = 0, and using Proposition 4.4.10 for the local error, we obtain en=

n 1

k=0

r(⌧A)^{n k} ¹ k+1

✓

(r(⌧A) e^⌧A)

n 1

k=0

r(⌧A)^{n k} ¹e^k⌧A

◆

A ¹(Du0+f)

= r(⌧A)ⁿ e^n⌧A A ¹(Du0+f).

Therefore, since A satisfies the hypotheses of Theorem 4.4.8, and since A ¹(Du0 +f) 2 D(A), we obtain the following estimate,

kenk^L^p^(⌦)  C tn

⌧²kDu0+fk^L^p^(⌦), which concludes the proof of Theorem 4.4.3.

Remark 4.4.12. Formula (4.22) can also be obtained directly with an appropriate change of variable. This alternative proof makes clear why the global error en can be expressed as a di↵erence between the rational function r(⌧A)ⁿ and the semigroup e^n⌧A. Indeed by the affine change of variable u(x, t) =ˆ u(x, t) z(x), whereˆ z(x) =ˆ z(x) A ¹Dz(x) A ¹f(x), we obtain @tu(x, t) =ˆ A˜u(x, t). Therefore, we can rewrite equation (4.17) as follows

@tu(x, t) =ˆ Aˆu(x, t) in ⌦⇥(0,1), Bu(x, t) = 0 onˆ @⌦⇥(0,1),

u(x,0) = u0(x) z(x) inˆ ⌦,

whose solution is given by u(x, t) = eˆ ^tAA ¹(Du0(x) +f(x)). Hence, we have the following formula for the exact solution,

u(x, t) = e^tAA ¹(Du0(x) +f(x)) + ˆz(x). (4.30) By Remark 4.4.2, we know that the numerical solution un of the splitting (4.6) is given by n iterations of the Crank-Nicolson method applied to the whole problem (4.17). Since Runge-Kutta methods are affine invariant, we obtain the following formula for u_n(x),

un(x) = r(⌧A)ⁿA ¹(Du0(x) +f(x)) + ˆz(x). (4.31) Subtracting (4.31) and (4.30) at time t_n, we therefore obtain formula (4.22) for the global error. Note that for any A-stable Runge-Kutta method with stability function R(y), ap-plied to the whole problem (4.17), the numerical solution un and the error en are given by formulas (4.31) and (4.22) with r(⌧A) replaced with R(⌧A).

Dans le document The Strang Splitting Method for Semilinear Parabolic Problems with Noncanonical Boundary Conditions (Page 67-73)