Continuous and Discontinuous Galerkin Time Stepping Methods for Nonlinear Initial Value Problems with Application to Finite Time Blow-Up
We consider continuous and discontinuous Galerkin time stepping methods of arbitrary order as applied to nonlinear initial value problems in real Hilbert spaces. Our only assumption is that the nonlinearities are continuous; in particular, we include the case of unbounded nonlinear operators. Specifically, we develop new techniques to prove general Peano-type existence results for discrete solutions. In particular, our results show that the existence of solutions is independent of the local approximation order, and only requires the local time steps to be sufficiently small (independent of the polynomial degree). The uniqueness of (local) solutions is addressed as well. In addition, our theory is applied to finite time blow-up problems with nonlinearities of algebraic growth. For such problems we develop a time step selection algorithm for the purpose of numerically computing the blow-up time, and provide a convergence result.
Key words and phrases:Initial value problems in Hilbert spaces, Galerkin time stepping schemes, high-order methods, blow-up singularities, existence and uniqueness of Galerkin solutions.
2010 Mathematics Subject Classification:65J08, 65L05, 65L60
In this paper we focus on continuous Galerkin (cG) as well as on discontinuous Galerkin (dG) time stepping discretizations (of any order) as applied to abstract initial value problems of the form
Here, , for some , is an unknown solution, with values in a real Hilbert space (with inner product denoted by and induced norm ). The initial value prescribes the solution at the start, , and is a possibly nonlinear, continuous operator. We emphasize that we include, for instance, the case of being (continuous and nonlinear and) unbounded in the sense that
In the sequel, we will usually omit to explicitly write the dependence on the first argument .
For and continuous nonlinearities , the well-known Peano Theorem (see, e.g., ) guarantees the existence of -solutions of (1.1) within some limited time range, , for some . Generalizations to problems in Banach spaces are available as well; see, e.g., . Notice that the existence interval for solutions may be arbitrarily small even for smooth : For instance, solutions of (1.1) may become unbounded in finite time, i.e.,
This effect is commonly termed (finite-time) blow-up.
Galerkin Time Stepping
Galerkin-type time stepping methods for initial-value problems are based on weak formulations. For both the cG and the dG time stepping schemes, the test spaces consist of polynomials that are discontinuous at the time nodes. In this way, the discrete Galerkin formulations decouple into local problems on each time step, and the discretizations can therefore be understood as implicit one-step schemes. Galerkin time stepping methods have been analyzed for ordinary differential equations (ODEs), e.g., in [3, 6, 5, 7, 8, 10].
A key feature of Galerkin time stepping methods is their great flexibility with respect to the size of the time steps and the local approximation orders, thereby naturally leading to an -version Galerkin framework. The -versions of the cG and dG time stepping schemes were introduced and analyzed in the works [12, 13, 15, 19]. In particular, in the articles [12, 19], which focus on ordinary initial value problems with uniform Lipschitz nonlinearities, the use of the contraction mapping theorem made it possible to prove existence and uniqueness results for discrete Galerkin solutions, which are independent of the local approximation orders. We emphasize that the -approach is well-known for its ability to approximate smooth solutions with possible local singularities at high algebraic or even exponential rates of convergence; see, e.g., [4, 13, 14, 18] for the numerical approximation of problems with start-up singularities.
The goal of the current paper is to extend the existence results on -type Galerkin time stepping schemes for initial value problems with Lipschitz-type nonlinearities in [12, 19] to problems with nonlinearities which are merely continuous. We emphasize that this generalization is substantial; indeed, it covers, for example, the case of unbounded nonlinearities as in (1.2). We will develop a new technique which is based on writing the weak Galerkin formulations in strong form along the lines of [1, 15]. Subsequently, suitable fixed-point forms will be derived. In the context of the cG method, this is accomplished within an integral equation framework. For the dG scheme, matters are more sophisticated, and a careful investigation of the discrete time derivative operator, which involves a lifting operator from , is required on the local polynomial approximation space; this operator turns out to be an isomorphism on the underlying polynomial spaces (with a continuity constant of the inverse operator that is independent of the local polynomial degrees) and allows to transform the strong dG form into a fixed point equation. For both the cG and the dG schemes the application of Brower’s fixed point theorem yields the existence of discrete solutions; see Theorem 1. In particular, as in the case of Lipschitz continuous nonlinearities [1, 15], the existence results do not depend on the local polynomial degrees, and only require the local time steps to be sufficiently small. In this sense, our theory constitutes a discrete version of Peano’s Theorem. Furthermore, employing a contraction argument along the lines of the approach presented in , we show that the local Galerkin formulations are uniquely solvable (within a certain range); cf. Theorem 2.
In addition, we apply our general theory to initial value problems with nonlinearities of algebraic growth, i.e., , with , , and for a given range of ; in this case, the initial value problem (1.1) features a solution that blows up in a finite time . We will show that a careful selection of locally varying time steps in the cG and dG time stepping schemes results in discrete solutions that blow up as well; in this context, we mention the paper  which illustrates the importance of variable step size selection. More precisely, following some ideas from , we derive an analysis which allows to choose the local time steps a posteriori as the time marching process is moving forward. We develop a time step selection algorithm which guarantees the existence and uniqueness of local solutions, and provides a numerical approximation of the exact blow-up time. Moreover, we prove a convergence result which shows that the blow-up time can be approximately arbitrarily well if the time steps are scaled sufficiently small.
The concepts and technical tools developed in our current work constitute an important stepping stone with regard to the numerical treatment of finite time blow-up problems in the context of nonlinear parabolic partial differential equations.
Our article is organized as follows: Section 2 presents the cG and dG time stepping schemes. Furthermore, Section 3 centres on the development of existence proofs for discrete solutions. The question of uniqueness is addressed in Section 4. Moreover, the application of our results to algebraically growing nonlinearities causing finite time blow-ups will be worked out in Section 5. Finally, the article closes with a few concluding remarks in Section 6.
Throughout the paper, Bochner spaces will be used: For an interval and a real Hilbert space as before, the space consists of all functions that are continuous on with values in . Moreover, introducing, for , the norm
we write to signify the space of measurable functions so that the corresponding norm is bounded. We notice that is a Hilbert space with inner product and induced norm
2. Galerkin Time Discretizations
In this section we present the -cG and -dG time stepping methods as applied to (1.1).
2.1. -cG Time Stepping
On an interval , , consider time nodes which introduce a time partition of into open time intervals , . The (possibly varying) length of a time interval is called the time step. Furthermore, to each interval we associate a polynomial degree which takes the role of a local approximation order. Moreover, given a (real) Hilbert space , an integer , and an interval , the set
signifies the space of all polynomials of degree at most on with values in .
In practical computations, the Hilbert space , on which (1.1) is based, will typically be replaced by a finite-dimensional subspace , , on each interval , . The -orthogonal projection from to is defined by
With these definitions, the (fully discrete) -cG time marching scheme is iteratively given as follows: For given initial value (with , where is the initial value from (1.1)), we find through the weak formulation
for any . Notice that, in order to enforce the initial condition on each individual time step, the local trial space has one degree of freedom more than the local test space. Furthermore, if , we remark that the continuous Galerkin solution is globally continuous on .
2.2. -dG Time Stepping
In order to define the discontinuous Galerkin scheme, some additional notation is required: We define the one-sided limits of a piecewise continuous function at each time node by
for any . We emphasize that, in contrast to the continuous Galerkin formulation, the trial and test spaces are the same for the discontinuous Galerkin scheme. This is due to the fact that the initial values are weakly imposed (by means of an upwind flux) on each time interval.
3. Existence of Discrete Solutions
Our general strategy of proof is to represent the Galerkin formulations in terms of strong equations, and then to derive suitable fixed-point formulations. Subsequently, the existence of discrete solutions will follow from the application of Brower’s fixed point theorem.
3.1. Existence of cG Solutions
We begin by rewriting (2.2) as finding such that
Here, denotes the -projection onto the space , which is uniquely defined by
Thence, noticing that , we obtain the strong form
Integration results in
We see that the operator
maps into itself, and hence, the integral equation (3.2) is a fixed point formulation,
Since is continuous, its maximum on the compact set ,
exists. We let
Then, we introduce
where , with .
Let be arbitrary, and such that
Then, using Bochner’s Theorem as well as the Cauchy-Schwarz inequality, yields
Taking into account the boundedness of the -projection on (with constant 1) leads to
Thus, we have , and more generally, it follows . Finally, since is convex and compact, and is continuous, Brower’s fixed point theorem implies that there exists at least one solution of (3.4) in , and thus of (2.2).
3.2. Existence of dG Solutions
The situation for the dG method is more involved. We will commence by looking at a discrete dG time operator appearing in the dG formulation.
3.2.1. Discrete dG Time Operator
Following [15, Section 4.1] we define the lifting operator, for ,
on a real Hilbert space , with inner product , and norm .
In view of this definition with , we have for the dG solution from (2.3):
Then, since , , all belong to , we arrive at the strong formulation
of (2.3). The term on the left-hand side of this equation is the -dG time discretization of the continuous derivative operator . This motivates the definition of a discrete operator
For the proof of existence of solutions of (3.8) it is important to notice that the linear operator is invertible.
In order to establish this estimate, we require two auxiliary results which will be proved first.
Let , and a real Hilbert space. Then, there holds
for any .
Let us first consider the lifting operator on the unit interval , defined by
Referring to [15, Eq. (35) and Lemma 8] there holds the explicit formula
with signifying the family of Legendre polynomials on (with degrees ), scaled such that ; cf. [15, Eq. (9) and Lemma 1]. Combining the above identities, we obtain
Noticing the telescope sum as well as the fact that and , we arrive at
Then, employing the fact that
Now we define the affine mapping
A scaling argument implies that
see [15, Lemma 7]. Hence, by a change of variables, , , we conclude that
Noticing that, for , there holds equality in the above bound, completes the proof. ∎
Let , and a real Hilbert space. Then, the bound
holds true for any .
Let . We define
Involving (3.13) shows
Furthermore, is orthogonal to the space (where ) with respect to the inner product in . In particular, since , we have
So, noticing that , it follows that
Therefore, using Hölder’s inequality and recalling (3.16), we conclude that
Dividing by shows the desired bound. ∎
We are now ready to show Proposition 1.
Proof of Proposition 1.
Upon setting in (3.11), we obtain
for any .
3.2.2. Fixed Point Formulation and Existence of Discrete dG Solutions
As for the cG method we prove the existence of solutions of (2.3) by means of a fixed point argument. For this purpose, we will derive a suitable fixed point formulation, and return to the case . Noticing the fact that , we observe that, on , there holds
and recalling (3.8), we can write
Applying Proposition 1 we infer that
this is the ‘dG-version’ of the integral equation (3.2) for the cG method. Now, for given (where as before ) we define the operator
Then, solves (3.8) if and only if satisfies
We will now prove the existence of solutions to the local -dG time stepping scheme (2.3): Consider (with ), and define the set
Due to the continuity of , its maximum on the compact set ,
exists. We choose
where is the constant from (3.11), and introduce
with , .