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 LieTrottersplitting and Strangsplitting schemes, see [21] and [20], which we call AB, ABA, BABschemes. 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].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 twosided onedimensional Wiener process. is the nonlinear fluxfunction, 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 nonequilibrium 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]:
The application of the separated solver methods is done with different splitting approaches, see an overview in [6].

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:

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 LieTrotter schemes, see [21] and [6], while we compute the numerical results for each operatorequation, see equation (8)(10) and couple the results as an initial value of the successor operatorequation, for example, we apply the results of equation (8) as an initial value for the equation (10), see also [6].

AB splitting:
We have the following AB splitting approaches:

APart
(12) where we have the solution
(13) 
Bpart:
(14) where we have the solution
(15) where we have the solution .


ABA splitting:
We have the following ABA splitting approaches:

APart ()
(16) where we have the solution
(17) 
Bpart:
(18) where we have the solution
(19) 
APart ()
(20) where we have the solution
(21) where we have the solution .


BAB splitting:
We have the following BAB splitting approach:

Bpart ():
(22) where we have the solution
(23) (24) 
APart ()
(25) where we have the solution
(26) 
Bpart:
(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 timeinterval the solver method and improve cyclic the solutions in this local timeinterval, 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 (Milsteinscheme) and space (finitevolume 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 fixpointscheme to improve the standard Milstein scheme (32) and obtain:
(35) 
where we have the initialization .
We deal with the following iterative splitting approaches:

Standard Milsteinscheme of second order ():
(36) 
Second order iterative splitting approach (related to the standard Milsteinscheme 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 ABsplitting approach, means with the rectangle rule and the semianalytical approach):

(Initialization):
(40) where .

(first step):
(41) where we apply the Ito’s rule with a first order scheme (EulerMaryamascheme) 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 .

We compute :
(44) we have as starting value.

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 ABAsplitting approach, means with the rectangle rule and the semianalytical approach):
The next algorithm for is given in 3.2, we improve the last with an underlying ABAmethod. We have to compute the solutions for .
Algorithm 3.2
We start with the initialization (initial value) and .

We compute :
(45) we have as starting value.

We compute (with ABA as solution for ):
(46) we have and obeys the Gaussian normal distribution with and .

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 timesteps which are computed by an underlying ABAmethod. We have to compute the solutions for .
Algorithm 3.3
We start with the initialization (initial value) and .

We compute :
(47) we have as starting value.

We compute (with ABA as solution for ):
(48) we have and and obeys the Gaussian normal distribution with and .

We obtain the next solution , If , we stop,
else we apply and goto step 1.
In figure 3, we see the further improvements of the iterative approaches.
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 Lipschitzconstant if
(51) 
for all .
We have the following assuptions:
Assumption 4.1
We have the Lipschitzconstinuous 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 Lipschitzconstants 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 TaylorIto scheme at the integration point and obtain, accuracy of the Milsteinscheme, which is given as:
(55)  
We apply:
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