Iterative and Non-iterative Splitting approach of a stochastic Burgers' equation

07/30/2019
by   Jürgen Geiser, et al.
Ruhr-Universität Bochum
0

In this paper we present iterative and noniterative splitting methods, which are used to solve stochastic Burgers' equations. The non-iterative splitting methods are based on Lie-Trotter and Strang-splitting methods, while the iterative splitting approaches are based on the exponential integrators for stochastic differential equations. Based on the nonlinearity of the Burgers' equation, we have investigated that the iterative schemes are more accurate and efficient as the non-iterative methods.

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1 Introduction

We are motivated to model the nonlinear transport phenomenon with stochastic perturbations. Such modelling problems arise in many fields, such as biology, physics, engineering and economics, where random phenomena play an important role. We concentrate on nonlinear transport with stochastic reaction, which can be modelled by the Burgers’ equation with an additional stochastic part. Many applications in transport phenomena can be modelled with uncertainties in combining deterministic and stochastic operators, see [14]. To solve such delicate problems, we consider operator splitting approaches to decompose into deterministic and stochastic operators, see [8] and [17]. The numerical schemes are discussed as noniterative schemes in the direction of Lie-Trotter-splitting and Strang-splitting schemes, see [21] and [20], which we call AB-, ABA-, BAB-schemes. The iterative schemes are discussed in the direction of Picard’s iterative schemes, see [6]. We apply the extension of the deterministic to the stochastic schemes, which are given in the numerical stochastics literature, e.g., [15], [18] and [5].

The benefit of splitting approaches arises in decomposing different operators, which can be solved numerically with more optimal methods. In the underlying stochastic Burgers’ equation, we decompose the deterministic part, which has to be solved with fast conservation methods, see [13] and [14]

, and the stochastic part, which has to be solved as a stochstatic ordinary differential equation, see

[13] and [14].

The paper is outlined as following: The model is given in section 2. The numerical methods are discussed in section 3. The numerical analysis is presented in section 4. In section 5, we present the numerical simulations. In the contents, that are given in section 6, we summarize our results.

2 Mathematical Model

In modeling, we concentrate on nonlinear stochastic PDEs (SPDEs), which are important to fluid dynamics. Here, we deal with stochastic Burgers equation (SBE) driven by linear multiplicative noise, see [9] and [12].

The SBE is given as:

(1)
(2)
(3)

where is a positive diffusion coefficient, is a two-sided one-dimensional Wiener process. is the nonlinear flux-function, e.g., . Further is globally Lipschitz continuous in . is multiplicative noise function and Lipschitz continuous in , which measures the amplitude of the noise. is an initial condition.

Such SPDEs driven by linear multiplicative noise and especially the SBE (10

) are used to model turbulences or non-equilibrium phase transitions, see

[4] and [16]. Further, the models are used to deal with randomly fluctuating environments [2] and also to model of parameter disturbances based on uncertainties, see [3].

We deal with a stochastic balance equation, which is given in the nonlinear transport case as a pure stochastic Burger’s equation:

(4)
(5)
(6)

where is the multiplicative noise function, is the nonlinear flux function and is a Wiener process.

In the next section, we deal with the different solver methods.

3 Numerical Methods

For the numerical methods, we apply based on the idea of separating the deterministic and stochastic operators, the following numerical approaches, see also [14, 13]:

  • Deterministic solver for the pure Burgers’ equation: Finite volume discretization for the space with the conservation law solver of Engquist-Osher, see [10] and [11].

  • Stochastic solver for the pure stochastic ODE: Euler-Maruyama solver, Milstein solver or stochastic RK solver, see [15] and [19].

The application of the separated solver methods is done with different splitting approaches, see an overview in [6].

We have the following parts of the full equation (10).

(7)

where

  • The deterministic part:

    (8)

    where we have the solution

    (9)
  • The stochastic part:

    (10)

    where we have the solution

    (11)

We concentrate on the following methods:

  • Noniterative methods based on exponential splitting approaches:

    • AB-splitting (Lie-Trotter scheme, see [14]),

    • ABA splitting (Strang-splitting scheme, see [17]),

    • BAB splitting (Strang-splitting scheme, see [17]),

  • Iterative method based on successive relaxation approaches:

    • Iterative splitting (Picard’s approximation, see [8]).

In the following, we discuss the different schemes.

3.1 Noniterative splitting approaches

The noniterative splitting approaches obtained results in one cycle, which means it is not necessary to relax the solution. We consider the ideas related to the exponential splitting based on the Lie-Trotter schemes, see [21] and [6], while we compute the numerical results for each operator-equation, see equation (8)-(10) and couple the results as an initial value of the successor operator-equation, for example, we apply the results of equation (8) as an initial value for the equation (10), see also [6].

  1. AB splitting:

    We have the following AB splitting approaches:

    • A-Part

      (12)

      where we have the solution

      (13)
    • B-part:

      (14)

      where we have the solution

      (15)

      where we have the solution .

  2. ABA splitting:

    We have the following ABA splitting approaches:

    • A-Part ()

      (16)

      where we have the solution

      (17)
    • B-part:

      (18)

      where we have the solution

      (19)
    • A-Part ()

      (20)

      where we have the solution

      (21)

      where we have the solution .

  3. BAB splitting:

    We have the following BAB splitting approach:

    • B-part ():

      (22)

      where we have the solution

      (23)
      (24)
    • A-Part ()

      (25)

      where we have the solution

      (26)
    • B-part:

      (27)

      where we have the solution

      (28)

      where we have the solution

      (29)

      where we have the solution .

3.2 Iterative splitting

The iterative splitting approaches are based on successive relaxation, means we apply several times in the same time-interval the solver method and improve cyclic the solutions in this local time-interval, see [6].

To apply the iterative approaches, we can apply the iterative solvers before or after a spatial discretization, means:

  • 1.) Iterative splitting after the discretization, we apply iterative schemes for the nonlinearities.

  • 2.) Iterative splitting before the discretization, we apply the iterative scheme to decompose the differential equation into a kernel and perturbation term.

3.2.1 Iterative scheme after discretization

We have the following SDE in continuous form:

(30)

and in the SDE form as:

(31)

We apply the discretization in time (Milstein-scheme) and space (finite-volume scheme) and obtain:

(32)

where we have the initialization .

Further the solution of the Burgers’ equation is given as:

(33)
(34)

while we apply for the linearization in the Burgers’ equation.

We apply a fixpoint-scheme to improve the standard Milstein scheme (32) and obtain:

(35)

where we have the initialization .

We deal with the following iterative splitting approaches:

  • Standard Milstein-scheme of second order ():

    (36)
  • Second order iterative splitting approach (related to the standard Milstein-scheme of second order) ():

    (37)

    where and

    obeys the Gaussian normal distribution

    with and .

3.2.2 Iterative scheme before the dicretization

We have the following iterative splitting approaches, before the discretization:

We have with:

(38)

where we have the initialization .

We have the solution

(39)

with initialization and .

We deal with the following iterative splitting approaches: First order iterative splitting approach (related to the AB-splitting approach, means with the rectangle rule and the semi-analytical approach):

  • (Initialization):

    (40)

    where .

  • (first step):

    (41)

    where we apply the Ito’s rule with a first order scheme (Euler-Maryama-scheme) and obtain:

    (42)

    where and obeys the Gaussian normal distribution with and .

    We improve the order to with the Milstein approach in the stochastic term and obtain:

    (43)

    and result to (while is linear and not dependent of , we only have to apply the derivative to ).

    The algorithm for is given in 3.1. We have to compute the solutions for .

    Algorithm 3.1

    We start with the initialization (initial value) and .

    1. We compute :

      (44)

      we have as starting value.

    2. We obtain the next solution , If , we stop,
      else we apply and goto step 1.

Second and third order iterative splitting approach (related to the ABA-splitting approach, means with the rectangle rule and the semi-analytical approach):

The next algorithm for is given in 3.2, we improve the last with an underlying ABA-method. We have to compute the solutions for .

Algorithm 3.2

We start with the initialization (initial value) and .

  1. We compute :

    (45)

    we have as starting value.

  2. We compute (with ABA as solution for ):

    (46)

    we have and obeys the Gaussian normal distribution with and .

  3. We obtain the next solution , If , we stop,
    else we apply and goto step 1.

The next algorithm for is given in 3.3, we improve the last with additional intermediate time-steps which are computed by an underlying ABA-method. We have to compute the solutions for .

Algorithm 3.3

We start with the initialization (initial value) and .

  1. We compute :

    (47)

    we have as starting value.

  2. We compute (with ABA as solution for ):

    (48)

    we have and and obeys the Gaussian normal distribution with and .

  3. We obtain the next solution , If , we stop,
    else we apply and goto step 1.

In figure 1, we have the improvement, which are done in Algorithm 3.2 and 3.3.

Figure 1: Function and visualization of the ABA-splitting approach for the Jacobian- (upper figure) and Gauss-Seidel-method (lower figure).

In figure 3, we see the further improvements of the iterative approaches.

Figure 2: Illustration how the iterative algorithms work in principle.
Figure 3: The improvements with the iterative approaches with the help of the AB- and ABA splitting approach.

4 Numerical Analysis

In the numerical analysis, we concentrate on the new iterative algorithms and present the approximation to the fixpoint of the solutions.

The iterative splitting scheme is given as:

(49)

where we have with the start condition .

We apply the integration and have the solution

(50)

with initialization and .

Definition 1

We have and . Further, is Lipschitz contiuous on with Lipschitz-constant if

(51)

for all .

We have the following assuptions:

Assumption 4.1

We have the Lipschitz-constinuous functions and , while we also assume is Lipschitz continuous.

Then, we have the following Lemma:

Lemma 1

We have and . Further, and are contraction mappings on , while we assume are Lipschitz continuous with constants and .

Proof

We have

(52)

While the operator for the pure deterministic Burgers’ equation is bounded with respect to and we obtain for sufficient small and .

Further, we have

(53)

while the operator is bounded and lipschitz continuous.

Theorem 4.2

is a closed subset of and and are contraction mappings on with Lipschitz-constants and , then the iterative scheme (4) converge linearly to with the factor .

Proof

We apply the iterative scheme:

(54)

and evalutate the integral based on the Taylor-Ito scheme at the integration point and obtain, accuracy of the Milstein-scheme, which is given as:

(55)

We apply: