This article proposes a new approach to the construction of a linearization method based on the iterative operator-splitting method for nonlinear differential equations. The convergence properties of such a method are studied. The main features of the proposed idea are the linearization of nonlinear equations and the application of iterative splitting methods. We present an iterative operator-splitting method with embedded Newton methods to solve nonlinearity. We confirm with numerical applications the effectiveness of the proposed iterative operator-splitting method in comparison with the classical Newton methods. We provide improved results and convergence rates.

1. Introduction

In this paper we propose a modified Jacobian-Newton iterative method to solve nonlinear differential equations. In the first paper we concentrate on ordinary differential equations, but numerical results are also obtained for partial differential equations. Basic studies of the operator-splitting methods are found in [1, 2]. Further important research was carried out to obtain a higher order for the splitting methods (see [3]). For this reason, the iterative splitting methods became more important for linear and nonlinear differential equations, while simple increasing of iteration steps affects the order of the scheme (see [4]). An interesting topic is Newton's methods for nonlinear problems with specifications for numerical implementations, (see [5]). Efficient modifications of Newton's methods, with regard to the computation of the Jacobian matrices are discussed in [6]. In our paper we discuss the benefit of the combination of splitting and linearization methods (see theoretical frameworks [7, 8]).

The outline of the paper is as follows. For our mathematical model we describe the convection-diffusion-reaction equation in Section 2. The fractional splitting is introduced in Section 3. We present the iterative splitting methods in Section 4. Section 5 discusses the Newton methods and their modifications. In Section 6 we present the numerical results from the solution of selected model problems. We end the article in Section 7 with a conclusion and comments.

2. Mathematical Model

The motivation for the study presented below originates from a computational simulation of heat-transfer [9] and convection-diffusion-reaction-equations [1013].

In the present paper we concentrate on ordinary differential equations, given as 𝜕𝑡𝑢(𝑡)=𝐴(𝑢(𝑡))𝑢(𝑡)+𝐵(𝑢(𝑡))𝑢(𝑡),𝑡(0,𝑇),(2.1)where the initial condition is 𝑢(0)=𝑢0. The operators 𝐴(𝑢) and 𝐵(𝑢) can be spatially discretized operators, that is, they can correspond to the discretized in space convection and diffusion operators (matrices). In the following, we deal with bounded nonlinear operators.

The aim of this paper is to present a new method based on Newton and iterative schemes.

In the next section we discuss the decoupling of the time-scale with a first-order fractional splitting method.

3. Fractional-Splitting Methods of First-Order for Linear Equations

First we describe the simplest operator-splitting, which is called sequential operator-splitting, for the following linear system of ordinary differential equations:𝜕𝑡𝑢(𝑡)=𝐴𝑢(𝑡)+𝐵𝑢(𝑡),𝑡(0,𝑇),(3.1)where the initial condition is 𝑢(0)=𝑢0. The operators 𝐴 and 𝐵 are linear and bounded operators in a Banach space (see also Section 2).

The sequential operator-splitting method is introduced as a method that solves two subproblems sequentially, where the different subproblems are connected via the initial conditions. This means that we replace the original problem (3.1) with the subproblems𝜕𝑢(𝑡)𝜕𝑡=𝐴𝑢(𝑡),with𝑢𝑡𝑛=𝑢𝑛,𝜕𝑢(𝑡)𝜕𝑡=𝐵𝑢(𝑡),with𝑢𝑡𝑛=𝑢𝑡𝑛+1,(3.2) where the splitting time-step is defined as 𝜏𝑛=𝑡𝑛+1𝑡𝑛. The approximated solution is 𝑢𝑛+1=𝑢(𝑡𝑛+1).

Clearly, the replacement of the original problem with the subproblems usually results in an error, called splitting error. The splitting error of the sequential operator-splitting method can be derived as (cf., e.g., [1, 2]).𝜌𝑛=1𝜏𝑛𝜏exp𝑛𝜏(𝐴+𝐵)exp𝑛𝐵𝜏exp𝑛𝐴𝑢𝑡𝑛=𝑂𝜏0,for[𝐴,𝐵]=0,𝑛,for[𝐴,𝐵]0,(3.3) where [𝐴,𝐵]=𝐴𝐵𝐵𝐴 is the commutator of 𝐴 and 𝐵. Consequently, the splitting error is 𝑂(𝜏𝑛) when the operators 𝐴 and 𝐵 do not commute, otherwise the method is exact. Hence, by definition, the sequential operator-splitting is called the first-order splitting method.

4. The Iterative Splitting Method

The following algorithm is based on the iteration with fixed splitting discretization step-size 𝜏. On the time interval [𝑡𝑛,𝑡𝑛+1] we solve the following subproblems consecutively for 𝑖=0,2,2𝑚:𝜕𝑢𝑖(𝑥,𝑡)𝜕𝑡=𝐴𝑢𝑖(𝑥,𝑡)+𝐵𝑢𝑖1(𝑥,𝑡),with𝑢𝑖(𝑡𝑛)=𝑢𝑛,𝑢0(𝑥,𝑡𝑛)=𝑢𝑛,𝑢1𝑢=0,𝑖(𝑥,𝑡)=𝑢𝑖1(𝑥,𝑡)=𝑢1,on𝜕Ω×(0,𝑇),𝜕𝑢𝑖+1(𝑥,𝑡)𝜕𝑡=𝐴𝑢𝑖(𝑥,𝑡)+𝐵𝑢𝑖+1(𝑥,𝑡),with𝑢𝑖+1(𝑥,𝑡𝑛)=𝑢𝑛,𝑢𝑖(𝑥,𝑡)=𝑢𝑖1(𝑥,𝑡)=𝑢1,on𝜕Ω×(0,𝑇),(4.1) where 𝑢𝑛 is the known split approximation at the time level 𝑡=𝑡𝑛 (see [14]).

Remark 4.1. We can generalize the iterative splitting method to a multi-iterative splitting method by introducing new splitting operators, for example, spatial operators. Then we obtain multi-indices to control the splitting process; each iterative splitting method can be solved independently, while connecting with further steps to the multi-splitting methods. In the following we introduce the multi-iterative splitting method for a combined time-space splitting method.

5. The Modified Jacobian-Newton Methods and Fixpoint-Iteration Methods

In this section we describe the modified Jacobian-Newton methods and Fixpoint-iteration methods.

We propose for weak nonlinearities, for example, quadratic nonlinearity, the fixpoint iteration method, where our iterative operator splitting method is one, see [4]. For stronger nonlinearities, for example, cubic or higher order polynomial nonlinearities, the modified Jacobian method with embedded iterative-splitting methods is suggested.

The point of embedding the splitting methods into the Newton methods is to decouple the equation systems into simpler equations. Such simple equation systems can be solved with scalar Newton methods.

5.1. The Altered Jacobian-Newton Iterative Methods with Embedded Sequential Splitting Methods

We confine our attention to time-dependent partial differential equations of the form𝑑𝑐𝑡𝑑𝑡=𝐴(𝑐(𝑡))𝑐(𝑡)+𝐵(𝑐(𝑡))𝑐(𝑡),with𝑐𝑛=𝑐𝑛,(5.1)where 𝐴(𝑐),𝐵(𝑐)𝐗𝐗 are linear and densely defined in the real Banach space 𝐗, involving only spatial derivatives of 𝑐, see [8]. We assume also that we have a weak nonlinear operator with 𝐴(𝑐)𝑐=𝜆1𝑐 and 𝐵(𝑐)𝑐=𝜆2𝑐, where 𝜆1 and 𝜆2 are constant factors.

In the following we discuss the embedding of a sequential splitting method into the Newton method.

The altered Jacobian-Newton iterative method with an embedded iterative splitting method is given as follows.

Newton's Method
𝐹(𝑐)=𝑑𝑐/𝑑𝑡𝐴(𝑐(𝑡))𝑐(𝑡)𝐵(𝑐(𝑡))𝑐(𝑡) and we can compute 𝑐(𝑘+1)=𝑐(𝑘)𝐷(𝐹(𝑐(𝑘)))1𝐹(𝑐(𝑘)), where 𝐷(𝐹(𝑐)) is the Jacobian matrix and 𝑘=0,1,.

We stop the iterations when we obtain: |𝑐(𝑘+1)𝑐(𝑘)|err, where err is an error bound, for example, err=104.

We assume the spatial discretization, with spatial grid points, 𝑖=1,,𝑚 and obtain the differential equation system:𝐹𝑐𝐹(𝑐)=1𝐹𝑐2𝐹𝑐𝑚,(5.2)where 𝑐=(𝑐1,,𝑐𝑚)𝑇 and 𝑚 is the number of spatial grid points.

The Jacobian matrix for the equation system is given as:𝐷𝐹(𝑐)=𝜕𝐹(𝑐1)𝑐1𝜕𝐹(𝑐1)𝑐2𝜕𝐹(𝑐1)𝑐𝑚𝜕𝐹(𝑐2)𝑐1𝜕𝐹(𝑐2)𝑐2𝜕𝐹(𝑐2)𝑐𝑚𝜕𝐹(𝑐𝑚)𝑐1𝜕𝐹(𝑐𝑚)𝑐2𝜕𝐹(𝑐𝑚)𝑐𝑚,(5.3)where 𝑐=(𝑐1,,𝑐𝑚).

The modified Jacobian is given as:𝐷𝐹(𝑐)=𝜕𝐹(𝑐1)𝑐1+𝐹(𝑐1)𝜕𝐹(𝑐1)𝑐2𝜕𝐹(𝑐1)𝑐𝑚𝜕𝐹(𝑐2)𝑐1𝜕𝐹(𝑐2)𝑐2𝐹(𝑐2)𝜕𝐹(𝑐2)𝑐𝑚𝜕𝐹(𝑐𝑚)𝑐1𝜕𝐹(𝑐𝑚)𝑐2𝜕𝐹(𝑐𝑚)𝑐𝑚+𝐹(𝑐𝑚),(5.4)where 𝑐=(𝑐1,,𝑐𝑛).

By embedding the sequential splitting method we obtain the following algorithm. We decouple into two equation systems:𝐹1𝑢1=𝜕𝑡𝑢1𝑢𝐴1𝑢1=0with𝑢1𝑡𝑛=𝑐𝑛,𝐹2𝑢2=𝜕𝑡𝑢2𝑢𝐵2𝑢2=0with𝑢2𝑡𝑛=𝑢1𝑡𝑛+1,(5.5)where the results of the methods are 𝑐(𝑡𝑛+1)=𝑢2(𝑡𝑛+1), and 𝑢1=(𝑢11,,𝑢1𝑛), 𝑢2=(𝑢21,,𝑢2𝑛).

Thus we have to solve two Newton methods, each in one equations system. The contribution is to reduce the Jacobian matrix into a diagonal entry, for example, with a weighted Newton method, see [15]. The splitting method with embedded Newton method is given as follows.

Algorithm 5.1. We assume the spatial operators 𝐴 and 𝐵 are discretized, for example, finite difference or finite element methods; further all initial conditions and boundary conditions are discrete given. Then we can apply the Newton's method in its discrete form as:𝑢1(𝑘+1)=𝑢1(𝑘)𝐹𝐷1𝑢1(𝑘)1𝜕𝑡𝑢1(𝑘)𝑢𝐴1(𝑘)𝑢1(𝑘),𝐹with𝐷1𝑢1(𝑘)=𝜕𝜕𝑢1(𝑘)𝜕𝑡𝑢1(𝑘)𝑢𝐴1(𝑘)𝑢𝜕𝐴1(𝑘)𝜕𝑢1(𝑘)𝑢1(𝑘),𝑢1(𝑘)𝑡𝑛=𝑐𝑛𝑢,𝑘=0,1,2,,𝐾,2(𝑙+1)=𝑢2(𝑙)𝐹𝐷2𝑢2(𝑙)1𝜕𝑡𝑢2(𝑙)𝑢𝐵2(𝑙)𝑢2(𝑙),𝐹with𝐷2𝑢2(𝑙)=𝜕𝜕𝑢1(𝑘)𝜕𝑡𝑢2(𝑘)𝑢𝐵2(𝑙)𝑢𝜕𝐵2(𝑙)𝜕𝑢2(𝑙)𝑢2(𝑙),𝑢2(𝑙)𝑡𝑛=𝑢𝐾1𝑡𝑛+1,𝑙=0,1,2,,𝐿.(5.6)where 𝑘 and 𝑙 are the iteration indices, 𝐾 and 𝐿 the maximal iterative steps for each part of the Newton's method. The maximal iterative steps allow us to have at least an error of: ||𝑢(𝐾)(𝑡𝑛+1)1𝑢(𝐾1)(𝑡𝑛+1)1||||𝑢err,(𝐿)(𝑡𝑛+1)2𝑢(𝐿1)(𝑡𝑛+1)2||err,(5.7) where err is the error bound, for example, err=106.
The approximated solution is given as: 𝑢𝑡𝑛+1=𝑢(𝐿)(𝑡𝑛+1)2.(5.8)

For the improvement method, we can apply the weighted Newton method. We try to skip the delicate outer diagonals in the Jacobian matrix and apply:𝑢1(𝑘+1)=𝑢1(𝑘)𝐷𝐹1𝑢1(𝑘)+𝛿1𝑢1(𝑘)1𝐹1𝑢1(𝑘)+𝜖𝑢1(𝑘),(5.8)where the function 𝛿 can be applied as a scalar, for example, 𝛿=106, and the same with 𝜖. It is important to ensure that 𝛿 is small enough to preserve the convergence.

Remark 5.2. If we assume that we discretize (5.5) with the backward-Euler method, for example,𝐹1𝑢1𝑡𝑛+1=𝑢1𝑡𝑛+1𝑢1𝑡𝑛𝑢Δ𝑡𝐴1𝑡𝑛+1𝑢1𝑡𝑛+1=0with𝑢1𝑡𝑛=𝑐𝑛,𝐹2𝑢2=𝑢2𝑡𝑛+1𝑢2𝑡𝑛𝑢Δ𝑡𝐵2𝑡𝑛+1𝑢2𝑡𝑛+1=0with𝑢2𝑡𝑛=𝑢1𝑡𝑛+1,(5.10) then we obtain the derivations 𝐷(𝐹1(𝑢1(𝑡𝑛+1))) and 𝐷(𝐹2(𝑢2(𝑡𝑛+1)))𝐷𝐹1𝑢1𝑡𝑛+1𝐴𝑢=1Δ𝑡1𝑡𝑛+1+𝑢𝜕𝐴1𝑡𝑛+1𝜕𝑢1𝑡𝑛+1𝑢1𝑡𝑛+1,𝐷𝐹2𝑢2𝐵𝑢=1Δ𝑡2𝑡𝑛+1+𝑢𝜕𝐵2𝑡𝑛+1𝜕𝑢2𝑡𝑛+1𝑢2𝑡𝑛+1.(5.11)

We can apply the equation (5.9) analogously 𝑢2(𝑙+1).

5.2. Iterative Operator-Splitting Method as a Fixpoint Scheme

The iterative operator-splitting method is used as a fixpoint scheme to linearize the nonlinear operators, see [4, 16].

We confine our attention to time-dependent partial differential equations of the form:𝑑𝑢𝑑𝑡=𝐴(𝑢(𝑡))𝑢(𝑡)+𝐵(𝑢(𝑡))𝑢(𝑡),with𝑢(𝑡𝑛)=𝑐𝑛,(5.12)where 𝐴(𝑢),𝐵(𝑢)𝐗𝐗 are linear and densely defined in the real Banach space 𝐗, involving only spatial derivatives of 𝑐, see [8]. In the following we discuss the standard iterative operator-splitting methods as a fixpoint iteration method to linearize the operators.

We split our nonlinear differential equation (5.12) by applying:𝑑𝑢𝑖(𝑡)𝑢𝑑𝑡=𝐴𝑖1𝑢(𝑡)𝑖𝑢(𝑡)+𝐵𝑖1𝑢(𝑡)𝑖1(𝑡),with𝑢𝑖𝑡𝑛=𝑐𝑛,𝑑𝑢𝑖+1(𝑡)𝑢𝑑𝑡=𝐴𝑖1𝑢(𝑡)𝑖𝑢(𝑡)+𝐵𝑖1𝑢(𝑡)𝑖+1,with𝑢𝑖+1𝑡𝑛=𝑐𝑛,(5.13)where the time-step is 𝜏=𝑡𝑛+1𝑡𝑛. The iterations are 𝑖=1,3,,2𝑚+1. 𝑢0(𝑡)=𝑐𝑛 is the starting solution, where we assume the solution 𝑐𝑛+1 is near 𝑐𝑛, or 𝑢0(𝑡)=0. So we have to solve the local fixpoint problem. 𝑐𝑛 is the known split approximation at the time level 𝑡=𝑡𝑛.

The split approximation at time level 𝑡=𝑡𝑛+1 is defined as 𝑐𝑛+1=𝑢2𝑚+2(𝑡𝑛+1). We assume the operators 𝐴(𝑢𝑖1),𝐵(𝑢𝑖1)𝐗𝐗 to be linear and densely defined on the real Banach space 𝐗, for 𝑖=1,3,,2𝑚+1.

Here the linearization is done with respect to the iterations, such that 𝐴(𝑢𝑖1),𝐵(𝑢𝑖1) are at least non-dependent operators in the iterative equations, and we can apply the linear theory.

The linearization is at least in the first equation 𝐴(𝑢𝑖1)𝐴(𝑢𝑖), and in the second equation 𝐵(𝑢𝑖1)𝐵(𝑢𝑖+1).

We have 𝐴𝑢𝑖1𝑡𝑛+1𝑢𝑖𝑡𝑛+1𝑢𝐴𝑛+1𝑢𝑡𝑛+1𝜖,(5.14)with sufficient iterations (5.14)𝑖={1,3,,2𝑚+1}.

RemarkThe linearization with the fixpoint scheme can be used for smooth or weak nonlinear operators, otherwise we lose the convergence behavior, while we did not converge to the local fixpoint, see [].

The second idea is based on the Newton method.

5.3. Jacobian-Newton Iterative Method with Embedded Operator-Splitting Method

The Newton method is used to solve the nonlinear parts of the iterative operator-splitting method (see the linearization techniques in [, ]).

Newton Method
The function is given as: 𝐹(𝑐)=𝜕𝑐𝜕𝑡𝐴(𝑐(𝑡))𝑐(𝑡)𝐵(𝑐(𝑡))𝑐(𝑡)=0,(5.14)
The iteration can be computed as: 𝑐(𝑘+1)=𝑐(𝑘)𝐹𝑐𝐷(𝑘)1𝐹𝑐(𝑘),(5.14) where (5.14)𝐷(𝐹(𝑐)) is the Jacobian matrix and (5.14)𝑘=0,1,. and (5.14)𝑐=(𝑐1,,𝑐𝑚) is the solution vector of the spatial discretized nonlinear equation.

We then have to apply the iterative operator-splitting method and obtain:𝐹1𝑢𝑖=𝜕𝑡𝑢𝑖𝑢𝐴𝑖𝑢𝑖𝑢𝐵𝑖1𝑢𝑖1=0,with𝑢𝑖𝑡𝑛=𝑐𝑛,𝐹2𝑢𝑖+2=𝜕𝑡𝑢𝑖+1𝑢𝐴𝑖𝑢𝑖𝑢𝐵𝑖+1𝑢𝑖+1=0,with𝑢𝑖+1𝑡𝑛=𝑐𝑛,(5.14)where the time-step is 𝜏=𝑡𝑛+1𝑡𝑛. The iterations are 𝑖=1,3,,2𝑚+1. 𝑐0(𝑡)=0 is the starting solution and 𝑐𝑛 is the known split approximation at the time-level 𝑡=𝑡𝑛. The results of the methods are 𝑐(𝑡𝑛+1)=𝑢2𝑚+2(𝑡𝑛+1).

Thus we have to solve two Newton methods and the contribution will be to reduce the Jacobian matrix, for example, skip the diagonal entries. The splitting method with the embedded Newton method is given as:𝑢𝑖(𝑘+1)=𝑢𝑖(𝑘)𝐹𝐷1𝑢𝑖(𝑘)1𝜕𝑡𝑢𝑖(𝑘)𝑢𝐴𝑖(𝑘)𝑢𝑖(𝑘)𝑢𝐵(𝑘)𝑖1𝑢(𝑘)𝑖1,𝐹with𝐷1𝑢𝑖(𝑘)𝐴𝑢=𝑖(𝑘)+𝑢𝜕𝐴𝑖(𝑘)𝜕𝑢𝑖(𝑘)𝑢𝑖(𝑘),𝑘=0,1,2,,𝐾,with𝑢𝑖𝑡𝑛=𝑐𝑛,𝑢(𝑙+1)𝑖+1=𝑢(𝑙)𝑖+1𝐹𝐷2𝑢(𝑙)𝑖+11𝜕𝑡𝑢(𝑙)𝑖+1𝑢𝐴𝑖(𝑘)𝑢𝑖(𝑘)𝑢𝐵(𝑘)𝑖+1𝑢(𝑘)𝑖+1𝑐2(𝑙),𝐹with𝐷2𝑢(𝑙)𝑖+1𝐵𝑢=(𝑙)𝑖+1+𝑢𝜕𝐵(𝑙)𝑖+1𝜕𝑢(𝑙)𝑖+1𝑢(𝑙)𝑖+1,𝑙=0,1,2,,𝐿,with𝑢𝑖+1𝑡𝑛=𝑐𝑛,(5.18)where the time-step is 𝜏=𝑡𝑛+1𝑡𝑛. The iterations are: 𝑖=1,3,,2𝑚+1. 𝑐0(𝑡)=0 is the starting solution and 𝑐𝑛 is the known split approximation at the time-level 𝑡=𝑡𝑛. The results of the methods are 𝑐(𝑡𝑛+1)=𝑢2𝑚+2(𝑡𝑛+1).

For the improvement by skipping the delicate outer diagonals in the Jacobian matrix, we apply 𝑢𝑖(𝑘+1)=𝑢𝑖(𝑘)(𝐷(𝐹1(𝑢𝑖(𝑘)))+𝛿1(𝑢𝑖(𝑘)))1(𝐹1(𝑢𝑖(𝑘))+𝜖𝑢𝑖(𝑘)), and analogously 𝑢(𝑙+1)𝑖+1.

Remark 5.4. For the iterative operator-splitting method with the Newton iteration we have two iteration procedures. The first iteration is the Newton method to compute the solution of the nonlinear equations, and the second iteration is the iterative splitting method, which computes the resulting solution of the coupled equation systems. The embedded method is used for strong nonlinearities.

6. Numerical Results

In this section, we present the numerical results for nonlinear differential equation using several variations of the proposed Newton and iterative schemes as solvers.

6.1. First Numerical Example: Bernoulli Equation

As a nonlinear differential example, we choose the Bernoulli equation:𝜕𝑢(𝑡)=𝜆𝜕𝑡1+𝜆3𝜆𝑢(𝑡)+2+𝜆4(𝑢(𝑡))𝑝,𝑡[0,𝑇],with𝑢(0)=1,(6.1)where the analytical solution can be derived as (see also [16]): 𝜆𝑢(𝑡)=exp1+𝜆3𝑡𝜆2+𝜆4𝜆1+𝜆3𝜆exp1+𝜆3(𝑝1)𝑡+𝑐1/(1𝑝).(6.2)

Using 𝑢(0)=1 we find that 𝑐=1+(𝜆2+𝜆4)/(𝜆1+𝜆3), so 𝜆𝑢(𝑡)=exp1+𝜆3𝑡𝜆1+2+𝜆4𝜆1+𝜆3𝜆1exp1+𝜆3(𝑝1)𝑡1/(1𝑝).(6.3)

We choose 𝑝=2, 𝜆1=1, 𝜆2=0.5, 𝜆3=100, 𝜆4=20 and, for example, Δ𝑡=102.

The analytical solutions can be given as:𝑢(𝑡)1𝑝=𝑢0𝜆exp(1𝑝)1+𝜆3𝑡+𝜆2+𝜆4𝜆1+𝜆3𝜆exp(1𝑝)1+𝜆3𝑡1).(6.4)

We divide the time interval [0,𝑇], with 𝑇=1, in 𝑛 intervals with length 𝜏𝑛=𝑇/𝑛.

(1) The sequential operator-splitting method with analytical solutions is given as follows.

We apply the quasilinear iterative operator-splitting method:𝑑𝑢1(𝑡)𝑢𝑑𝑡=𝐴1𝑢(𝑡)1(𝑡),with𝑢1𝑡𝑛=𝑢𝑛,𝑑𝑢2(𝑡)𝑢𝑑𝑡=𝐵2𝑢(𝑡)2,with𝑢2𝑡𝑛=𝑢1𝑛+1,(6.5) with the nonlinear operators 𝐴(𝑢)𝑢=𝜆1𝑢(𝑡)+𝜆2(𝑢(𝑡))𝑝1𝑢,𝐵(𝑢)𝑢=𝜆3𝑢(𝑡)+𝜆4(𝑢(𝑡))𝑝1𝑢. The result is given as 𝑢2(𝑡𝑛+1)=𝑢𝑛+1.

We apply the Newton method and discretize the operators with time discretization methods such as backward-Euler or higher Runge-Kutta methods.

The analytical result for each equation part is given as:𝑢1(𝑡)1𝑝𝑡=𝑢𝑛𝜆exp(1𝑝)1𝑡+𝜆2𝜆1𝜆exp(1𝑝)1𝑡,𝑢12𝑡𝑛+11𝑝=𝑢1𝑡𝑛+1𝜆exp(1𝑝)3𝑡+𝜆4𝜆3𝜆exp(1𝑝)3𝑡,1(6.6) where the result is given as 𝑢(𝑡𝑛+1)=𝑢2(𝑡𝑛+1).

We can apply the simpler equations and solve the sequential operator-splitting method.

(2) The sequential operator-splitting method with embedded Newton method is given as follows.

We apply the quasilinear iterative operator-splitting method:𝑑𝑢1(𝑡)𝑢𝑑𝑡=𝐴1𝑢(𝑡)1(𝑡),with𝑢1𝑡𝑛=𝑢𝑛,𝑑𝑢2(𝑡)𝑢𝑑𝑡=𝐵2𝑢(𝑡)2,with𝑢2𝑡𝑛=𝑢1𝑛+1,(6.7) with the nonlinear operators 𝐴(𝑢)𝑢=𝜆1𝑢(𝑡)+𝜆2(𝑢(𝑡))𝑝1𝑢,𝐵(𝑢)𝑢=𝜆3𝑢(𝑡)+𝜆4(𝑢(𝑡))𝑝1𝑢. The result is given as 𝑢2(𝑡𝑛+1)=𝑢𝑛+1.

We apply the Newton method and discretize the operators with time discretization methods such as backward-Euler or higher Runge-Kutta methods.

The splitting method with embedded Newton's method is given as𝑢1(𝑘+1)=𝑢1(𝑘)𝐹𝐷1𝑢1(𝑘)1𝜕𝑡𝑢1(𝑘)𝑢𝐴1(𝑘)𝑢1(𝑘),𝐹with𝐷1𝑢1(𝑘)𝐴𝑢=1(𝑘)+𝑢𝜕𝐴1(𝑘)𝜕𝑢1(𝑘)𝑢1(𝑘),𝑢1(𝑘)𝑡𝑛=𝑐𝑛𝑢,𝑘=0,1,2,,𝐾,2(𝑙+1)=𝑢2(𝑙)𝐹𝐷2𝑢2(𝑙)1𝜕𝑡𝑢2(𝑙)𝑢𝐵2(𝑙)𝑢2(𝑙),𝐹with𝐷2𝑢2(𝑙)𝐵𝑢=2(𝑙)+𝑢𝜕𝐵2(𝑙)𝜕𝑢2(𝑙)𝑢2(𝑙),𝑢2(𝑙)𝑡𝑛=𝑢𝐾1𝑡𝑛+1,𝑙=0,1,2,,𝐿,(6.8) where we discretize the equations and obtain the discretized operators:𝜕𝑡𝑢1(𝑘)𝑢𝐴1(𝑘)𝑢1(𝑘)=0,(6.9)as𝐹1𝑢1𝑡𝑛+1𝑢1(𝑘)𝑡𝑛+1𝑢1𝑡𝑛𝑢Δ𝑡𝐴1(𝑘)𝑡𝑛+1𝑢1(𝑘)𝑡𝑛+1=0,(6.10)where we have the initialization of the Newton's method as 𝑢1(0)(𝑡𝑛+1)=0 or 𝑢1(0)(𝑡𝑛+1)=𝑢1(𝑡𝑛).

For the second iteration equation we have𝜕𝑡𝑢2(𝑙)𝑢𝐵2(𝑙)𝑢2(𝑙),(6.11)as𝐹2𝑢2𝑡𝑛+1=𝑢2(𝑙)𝑡𝑛+1𝑢2𝑡𝑛𝑢Δ𝑡𝐵2(𝑙)𝑡𝑛+1𝑢2(𝑙)𝑡𝑛+1=0,(6.12)where we have the initialization of the Newton's method as 𝑢2(0)(𝑡𝑛+1)=0 or 𝑢2(0)(𝑡𝑛+1)=𝑢1(𝑡𝑛).

The derivations are given as:𝐷𝐹1𝑢1𝑡𝑛+1𝐴𝑢=1Δ𝑡1𝑡𝑛+1+𝑢𝜕𝐴1𝑡𝑛+1𝜕𝑢1𝑡𝑛+1𝑢1𝑡𝑛+1,𝐷𝐹2𝑢2𝐵𝑢=1Δ𝑡2𝑡𝑛+1+𝑢𝜕𝐵2𝑡𝑛+1𝜕𝑢2𝑡𝑛+1𝑢2𝑡𝑛+1.(6.13)

(3) The standard iterative operator-splitting method is given as follows.

We apply the quasilinear iterative operator-splitting method:𝑑𝑢𝑖(𝑡)𝑢𝑑𝑡=𝐴𝑖1𝑢(𝑡)𝑖𝑢(𝑡)+𝐵𝑖1𝑢(𝑡)𝑖1(𝑡),with𝑢𝑖𝑡𝑛=𝑢𝑛,𝑑𝑢𝑖+1(𝑡)𝑢𝑑𝑡=𝐴𝑖1𝑢(𝑡)𝑖𝑢(𝑡)+𝐵𝑖1𝑢(𝑡)𝑖+1,with𝑢𝑖+1𝑡𝑛=𝑢𝑛,(6.14) with the nonlinear operators 𝐴(𝑢)𝑢=𝜆1𝑢(𝑡)+𝜆2(𝑢(𝑡))𝑝1𝑢,𝐵(𝑢)𝑢=𝜆3𝑢(𝑡)+𝜆4(𝑢(𝑡))𝑝1𝑢. The initialization of the fixpoint iteration is 𝑢0=𝑢𝑛 or 𝑢0=0 with 𝐴(𝑢0)=𝜆1 and 𝐵(𝑢0)=𝜆3.

For the iterations we can apply the analytical solution of each equation:𝑢𝑖(𝑡)=𝑢𝑛𝐴𝑢exp𝑖1𝑡+𝐴𝑢(𝑡)𝑖1(𝑡)1𝐵𝑢𝑖1𝑢(𝑡)𝑖1𝐴𝑢(𝑡)1exp𝑖1𝑡,𝑢(𝑡)𝑖+1(𝑡)=𝑢𝑛𝐵𝑢exp𝑖1𝑡+𝐵𝑢(𝑡)𝑖1(𝑡)1𝐴(𝑢𝑖1𝑢(𝑡)𝑖1𝐵𝑢(𝑡)1exp𝑖1𝑡.(𝑡)(6.15)

Further the iterative steps can be done.

(4) The Newton iterative method with embedded iterative operator-splitting method is given as follows.

We apply the quasilinear iterative operator-splitting method:𝑑𝑢𝑖(𝑡)𝑢𝑑𝑡=𝐴𝑖𝑢(𝑡)𝑖𝑢(𝑡)+𝐵𝑖1𝑢(𝑡)𝑖1(𝑡),with𝑢𝑖𝑡𝑛=𝑢𝑛,𝑑𝑢𝑖+1(𝑡)𝑢𝑑𝑡=𝐴𝑖𝑢(𝑡)𝑖𝑢(𝑡)+𝐵𝑖+1𝑢(𝑡)𝑖+1,with𝑢𝑖+1𝑡𝑛=𝑢𝑛,(6.16)with the nonlinear operators 𝐴(𝑢)𝑢=𝜆1𝑢(𝑡)+𝜆2(𝑢(𝑡))𝑝1𝑢,𝐵(𝑢)𝑢=𝜆3𝑢(𝑡)+𝜆4(𝑢(𝑡))𝑝1𝑢. The initialization of the fixpoint iteration is 𝑢0(𝑡𝑛+1)=𝑢𝑛 or 𝑢0(𝑡𝑛+1)=0.

The discretization of the nonlinear ordinary differential equation is performed with higher-order Runge-Kutta methods.

The Newton method is applied as:𝑢𝑖(𝑘+1)=𝑢𝑖(𝑘)𝐹𝐷1𝑢𝑖(𝑘)1𝜕𝑡𝑢𝑖(𝑘)𝑢𝐴𝑖(𝑘)𝑢𝑖(𝑘)𝑢𝐵𝑖1𝑢𝑖1,𝐷𝐹1𝑢𝑖(𝑘)𝐴𝑢=𝑖(𝑘)+𝑢𝜕𝐴𝑖(𝑘)𝜕𝑢𝑖(𝑘)𝑢𝑖(𝑘)𝑢,𝑘=0,1,2,,𝐾,𝑖𝑡𝑛=𝑐𝑛,𝑢𝑖𝑡𝑛+1=𝑢𝑖𝑡𝑛+1𝐾+1,where|𝑢𝑖𝑡𝑛+1𝐾+1𝑢𝑖𝑡𝑛+1𝐾𝑢|err,(𝑙+1)𝑖+1=𝑢(𝑙)𝑖+1𝐹𝐷2𝑢(𝑙)𝑖+11𝜕𝑡𝑢(𝑙)𝑖+1𝑢𝐴𝑖𝑢𝑖𝑢𝐵(𝑙)𝑖+1𝑢(𝑙)𝑖+1𝑐2(𝑙),𝐷𝐹2𝑢(𝑙)𝑖+1𝐵𝑢=(𝑙)𝑖+1+𝑢𝜕𝐵(𝑙)𝑖+1𝜕𝑢(𝑙)𝑖+1𝑢(𝑙)𝑖+1𝑢,𝑙=0,1,2,,𝐿,𝑖+1𝑡𝑛=𝑐𝑛,𝑢𝑖+1𝑡𝑛+1=𝑢𝑖+1𝑡𝑛+1𝐿+1,where|𝑢𝑖+1𝑡𝑛+1𝐿+1𝑢𝑖+1𝑡𝑛+1𝐿|err.(6.17)Here the time-step is 𝜏=𝑡𝑛+1𝑡𝑛. The iterations are 𝑖=1,3,,2𝑚+1. 𝑢0(𝑡)=0 is the starting solution and 𝑐𝑛 is the known split approximation at the time-level 𝑡=𝑡𝑛. The results of the methods are 𝑢(𝑡𝑛+1)=𝑢2𝑚+2(𝑡𝑛+1).

We apply the discretization methods for the iteration steps.

We discretize the equations𝜕𝑡𝑢𝑖(𝑘)𝑢𝐴𝑖(𝑘)𝑢𝑖(𝑘)𝑢𝐵𝑖1𝑢𝑖1=0,(6.18)as𝐹1𝑢𝑖(𝑘)𝑡𝑛+1=𝑢𝑖(𝑘)𝑡𝑛+1𝑢𝑖𝑡𝑛𝐴𝑢Δ𝑡𝑖(𝑘)𝑡𝑛+1𝑢𝑖(𝑘)𝑡𝑛+1𝑢+𝐵𝑖1𝑡𝑛+1𝑢𝑖1𝑡𝑛+1,(6.19)where we have the initialization of the Newton's method as 𝑢𝑖(0)(𝑡𝑛+1)=0 or 𝑢𝑖(0)(𝑡𝑛+1)=𝑢(𝑡𝑛).

For the second iteration equation we have:𝜕𝑡𝑢2(𝑙)𝑢𝐴𝑖𝑢𝑖𝑢𝐵2(𝑙)𝑢2(𝑙)=0,(6.20)as𝐹2𝑢(𝑙)𝑖+1𝑡𝑛+1=𝑢(𝑙)𝑖+1𝑡𝑛+1𝑢𝑖+1𝑡𝑛𝐴𝑢Δ𝑡𝑖𝑡𝑛+1𝑢𝑖𝑡𝑛+1𝑢+𝐵(𝑙)𝑖+1𝑡𝑛+1𝑢(𝑙)𝑖+1𝑡𝑛+1,(6.21)where we have the initialization of the Newton's method as 𝑢(0)𝑖+1(𝑡𝑛+1)=0 or 𝑢(0)𝑖+1(𝑡𝑛+1)=𝑢(𝑡𝑛).

The derivations are given as:𝐷𝐹1𝑢𝑖(𝑘)𝑡𝑛+1𝐴𝑢=1Δ𝑡(𝑘)𝑖+1𝑡𝑛+1+𝑢𝜕𝐴(𝑘)𝑖+1𝑡𝑛+1𝜕𝑢(𝑘)𝑖+1𝑡𝑛+1𝑢(𝑘)𝑖+1𝑡𝑛+1,𝐷𝐹2𝑢(𝑙)𝑖+1𝑡𝑛+1𝐵𝑢=1Δ𝑡(𝑙)𝑖+1𝑡𝑛+1+𝑢𝜕𝐵(𝑙)𝑖+1𝑡𝑛+1𝜕𝑢(𝑙)𝑖+1𝑡𝑛+1𝑢(𝑙)𝑖+1𝑡𝑛+1.(6.22)

Our numerical results for the different methods are presented in Tables 1, 2, 3, and 4. The errors of the methods are shown in Figures 1, 2, and 3. We chose different iteration steps and time partitions. The error between the analytical and numerical solution is shown with the supremum norm at time 𝑇=1.0.

The experiments show the reduced errors for more iteration steps and more time partitions. Because of the time-discretization method for ODEs, we restrict the number of iteration steps to a maximum of five. If we restrict the error bound to 103, two iteration steps and five time partitions give the most effective combination.

6.2. Second Numerical Example: Mixed Convection-Diffusion and Burgers Equation

We deal with a 2D example which is a mixture of a convection-diffusion and Burgers equation. We can derive an analytical solution:𝜕𝑡1𝑢=2𝑢𝜕𝑥1𝑢2𝑢𝜕𝑦1𝑢2𝜕𝑥1𝑢2𝜕𝑦𝑢𝜕+𝜇𝑥𝑥𝑢+𝜕𝑦𝑦𝑢+𝑓(𝑥,𝑦,𝑡),(𝑥,𝑦,𝑡)Ω×[0,𝑇],𝑢(𝑥,𝑦,0)=𝑢ana(𝑥,𝑦,0),(𝑥,𝑦)Ω,𝑢(𝑥,𝑦,𝑡)=𝑢ana(𝑥,𝑦,𝑡)on𝜕Ω×[0,𝑇],(6.23)where Ω=[0,1]×[0,1], 𝑇=1.25, and 𝜇 is the viscosity.

The analytical solution is given as𝑢ana(𝑥,𝑦,𝑡)=1+exp𝑥+𝑦𝑡2𝜇1+exp𝑥+𝑦𝑡2𝜇,(6.24)where we compute 𝑓(𝑥,𝑦,𝑡) accordingly.

We split the convection-diffusion and the Burgers equation. The operators are given as: 1𝐴(𝑢)𝑢=2𝑢𝜕𝑥1𝑢2𝑢𝜕𝑦1𝑢+2𝜇𝜕𝑥𝑥𝑢+𝜕𝑦𝑦𝑢,(6.25) hence 1𝐴(𝑢)=2𝑢𝜕𝑥𝑢𝜕𝑦𝜕+𝜇𝑥𝑥+𝜕𝑦𝑦1(theBurgersterm),𝐵𝑢=2𝜕𝑥1𝑢2𝜕𝑦1𝑢+2𝜇𝜕𝑥𝑥𝑢+𝜕𝑦𝑦𝑢+𝑓(𝑥,𝑦,𝑡)(theconvection-diusionterm).(6.26)

For the first equation we apply the nonlinear Algorithm 5.1 and obtain 𝐴𝑢𝑖1𝑢𝑖1=2𝑢𝑖1𝜕𝑥𝑢𝑖12𝑢𝑖1𝜕𝑦𝑢𝑖+12𝜇𝜕𝑥𝑥𝑢𝑖+𝜕𝑦𝑦𝑢𝑖,𝐵𝑢𝑖1=12𝜕𝑥𝜕𝑦𝜕+𝜇𝑥𝑥+𝜕𝑦𝑦𝑢𝑖1,(6.27)and we obtain linear operators, because 𝑢𝑖1 is known from the previous time-step.

In the second equation we obtain by using Algorithm 5.1: 𝐴𝑢𝑖1𝑢𝑖1=2𝑢𝑖1𝜕𝑥𝑢𝑖12𝑢𝑖1𝜕𝑦𝑢𝑖+12𝜇𝜕𝑥𝑥𝑢𝑖+𝜕𝑦𝑦𝑢𝑖,𝐵𝑢𝑖+1=12𝜕𝑥𝜕𝑦𝜕+𝜇𝑥𝑥+𝜕𝑦𝑦𝑢𝑖+1,(6.28)and we have linear operators.

We deal with different viscosities 𝜇 as well as different step-sizes in time and space. We have the following results (see Tables 5 and 6).

Figure 4 presents the profile of the 2D linear and nonlinear convection-diffusion equation.

Remark 6.1. In the examples, we deal with more iteration steps to obtain higher-order convergence results. In the first test we have four iterative steps but a smaller viscosity (𝜇=0.5) such that we can reach at least a second-order method. In the second test we use a high viscosity about 𝜇=5 and get the second-order result with two iteration steps. Here we see the loss of differentiability, that becomes stiff equation parts. To obtain the same results, we have to increase the number of iteration steps. Therefore we can show an improvement of the convergence order with respect to the iteration steps.

6.3. Third Numerical Example: Momentum Equation (Molecular Flow)

We deal with an example of a momentum equation, that is used to model the viscous flow of a fluid.𝜕𝑡1𝐮=𝐮𝐮+2𝜇𝐷(𝐮)+3𝐮+𝐟(𝑥,𝑦,𝑡),(𝑥,𝑦,𝑡)Ω×[0,𝑇],𝐮(𝑥,𝑦,0)=𝐠1(𝑥,𝑦),(𝑥,𝑦)Ω,𝐮(𝑥,𝑦,𝑡)=𝐠2(𝑥,𝑦,𝑡)on𝜕Ω×[0,𝑇],(enclosedow),(6.29)where 𝐮=(𝑢1,𝑢2)𝑡 is the solution and Ω=[0,1]×[0,1], 𝑇=1.25, 𝜇=5, and 𝐯=(0.001,0.001)𝑡 are the parameters and 𝐼 is the unit matrix.

The nonlinear function 𝐷(𝐮)=𝐮𝐮+𝐯𝐮 is the viscosity flow, and 𝐯 is a constant velocity.

We can derive the analytical solution with respect to the first two test examples with the functions:𝑢1(𝑥,𝑦,𝑡)=1+exp𝑥+𝑦𝑡2𝜇1+exp𝑥+𝑦𝑡,𝑢2𝜇2(𝑥,𝑦,𝑡)=1+exp𝑥+𝑦𝑡2𝜇1+exp𝑥+𝑦𝑡.2𝜇(6.30) For the splitting method our operators are given as: 2𝐴(𝐮)𝐮=𝐮𝐮+2𝜇𝐷(𝐮)(thenonlinearoperator),𝐵𝐮=3𝜇Δ𝐮(thelinearoperator).(6.31)

We deal first with the one-dimensional case,𝜕𝑡𝑢=𝑢𝜕𝑥𝑢+2𝜇𝜕𝑥1𝐷(𝑢)+3𝜕𝑥𝑢+𝑓(𝑥,𝑡),(𝑥,𝑡)Ω×[0,𝑇],𝑢(𝑥,0)=𝑔1(𝑥),(𝑥)Ω,𝑢(𝑥,𝑡)=𝑔2(𝑥,𝑡)on𝜕Ω×[0,𝑇],(enclosedow),(6.32)where 𝑢 is the solution and Ω=[0,1], 𝑇=1.25, 𝜇=5, and 𝑣=0.001 are the parameters.

Then the operators are given as: 𝐴(𝑢)𝑢=𝑢𝜕𝑥𝑢+2𝜇𝜕𝑥2𝐷(𝑢)(thenonlinearoperator),𝐵𝑢=3𝜇𝜕𝑥𝑥𝑢(thelinearoperator).(6.33) For the iterative operator-splitting as fixed point scheme, we have the following results (see Tables 7 and 9).

Figure 5 presents the profile of the 1D momentum equation.

We have the following results for the 2D case (see Tables 10, 11, and 12).

Figure 6 presents the profile of the 2D momentum equation.

For the Newton operator-splitting method we obtain the following functional matrices for the one-dimensional case:𝐷𝐹(𝑢)=(4𝜇1)𝜕𝑥𝜕𝑢,𝐷(𝐹(𝑢))=𝑢1𝐹1(𝑢)𝜕𝑢2𝐹1𝜕(𝑢)𝑢1𝐹2(𝑢)𝜕𝑢2𝐹2(𝑢)=𝜕𝑥𝑢1+4𝜇𝜕𝑥𝑢1𝜕𝑥𝑢2+4𝜇𝜕𝑥𝑢2𝜕𝑦𝑢1+4𝜇𝜕𝑦𝑢1𝜕𝑦𝑢2+4𝜇𝜕𝑦𝑢2=(4𝜇1)𝑢.(6.34)For the two-dimensional case, we use:𝑢𝐴(𝐮)𝐮=𝐮𝐮+2𝜇𝐷(𝐮)=1𝜕𝑥𝑢1+𝑢2𝜕𝑥𝑢2𝑢1𝜕𝑦𝑢1+𝑢2𝜕𝑦𝑢2+2𝜇2𝑢1𝜕𝑥𝑢1+2𝑢2𝜕𝑥𝑢2+𝑣1𝜕𝑥𝑢1+𝑣2𝜕𝑥𝑢22𝑢1𝜕𝑦𝑢1+2𝑢2𝜕𝑦𝑢2+𝑣1𝜕𝑦𝑢1+𝑣2𝜕𝑦𝑢2.(6.35)

Here, we do not need the linearization and apply the standard iterative splitting method.

We only linearize the first split step and therefore we can relax this step with the second linear split step. Therefore we obtain stable methods, see [15]. For the Newton iterative method, we have the following results, see Table 8.

Remark 6.2. In the more realistic examples of 1D and 2D momentum equations, we can also observe the stiffness problem, which we obtain with a more hyperbolic behavior. In the 1D experiments we deal with a more hyperbolic behavior and can obtain at least first-order convergence with two iterative steps. In the 2D experiments we obtain nearly second-order convergence results with two iterative steps, if we increase the parabolic behavior, for example, larger 𝜇 and 𝐯 values. For such methods, we have to balance the usage of the iterative steps with refinement in time and space with respect to the hyperbolicity of the equations. At least we can obtain a second-order method with more than two iterative steps. Therefore the stiffness influences the number of iterative steps.

7. Conclusion and Discussion

We present decomposition methods for differential equations based on iterative and non-iterative methods. The nonlinear equations are solved with embedded Newton's methods. We present new ideas on linearization to obtain more accurate results. The superiority of the new embedded Newton's methods over the traditional sequential methods is demonstrated in examples, especially through their simple implementation. Further, we have the smoothing properties of the iterative scheme that allow a balance between the nonlinear and the linear terms. The results show more accurate solutions with respect to time decomposition. In the future the iterative operator-splitting method can be generalized for multi-dimensional problems and also for non-smooth and nonlinear problems in time and space. In next paper we discuss error analysis of nonlinear methods.