# Modified Jacobian Newton Iterative Method: Theory and Applications

Authors: Jürgen Geiser
Journal: Mathematical Problems in Engineering
 (2009)

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## Abstract

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.


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## Body

## 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
Section2. 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 [10–13].In the present paper we concentrate on ordinary
differential equations, given as(2.1)∂tu(t)=A(u(t))u(t)+B(u(t))u(t),t∈(0,T),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 calledsequential operator-splitting, for the
following linear system of ordinary differential equations:(3.1)∂tu(t)=Au(t)+Bu(t),t∈(0,T),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(3.2)∂u*(t)∂t=Au*(t),withu*(tn)=un,∂u**(t)∂t=Bu**(t),withu**(tn)=u*(tn+1),  where the splitting time-step is
defined as τn=tn+1−tn.
The approximated solution is un+1=u**(tn+1).Clearly, the replacement of the original problem with
the subproblems usually results in an error, calledsplitting error. The splitting error of
the  sequential operator-splitting  method can be derived as (cf., e.g., [1, 2]).(3.3)ρn=1τn(exp(τn(A+B))−exp(τnB)exp(τnA))u(tn)={0,for[A,B]=0,O(τn),for[A,B]≠0,  where [A,B]:=AB−BA 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-sizeτ.
On the time interval [tn,tn+1] we solve the following subproblems consecutively
for i=0,2,…2m:(4.1)∂ui(x,t)∂t=Aui(x,t)+Bui−1(x,t),withui(tn)=un,u0(x,tn)=un,u−1=0,ui(x,t)=ui−1(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)=ui−1(x,t)=u1,on∂Ω×(0,T), 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 form(5.1)dcdt=A(c(t))c(t)+B(c(t))c(t),withc(tn)=cn,where A(c),B(c):X→X 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/dt−A(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=10−4.We assume the spatial discretization, with spatial
grid points,i=1,…,m and obtain the differential equation
system:(5.2)F(c)=(F(c1)F(c2)⋮F(cm)),where c=(c1,…,cm)T and m is the number of spatial grid points.The Jacobian matrix for the equation system is given
as:(5.3)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),where c=(c1,…,cm).The modified Jacobian is given as:(5.4)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)),where c=(c1,…,cn).By embedding the sequential splitting method we obtain
the following algorithm.  We decouple
into two equation systems:(5.5)F1(u1)=∂tu1−A(u1)u1=0withu1(tn)=cn,F2(u2)=∂tu2−B(u2)u2=0withu2(tn)=u1(tn+1),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 operatorsA 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:(5.6)u1(k+1)=u1(k)−D(F1(u1(k)))−1(∂tu1(k)−A(u1(k))u1(k)),withD(F1(u1(k)))=∂∂u1(k)(∂tu1(k)−A(u1(k))−∂A(u1(k))∂u1(k)u1(k)),u1(k)(tn)=cn,k=0,1,2,…,K,u2(l+1)=u2(l)−D(F2(u2(l)))−1(∂tu2(l)−B(u2(l))u2(l)),withD(F2(u2(l)))=∂∂u1(k)(∂tu2(k)−B(u2(l))−∂B(u2(l))∂u2(l)u2(l)),u2(l)(tn)=u1K(tn+1),l=0,1,2,…,L.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:
(5.7)|u1(K)(tn+1)−u1(K−1)(tn+1)|≤err,|u2(L)(tn+1)−u2(L−1)(tn+1)|≤err,
where err is the error bound, for example, er r=10−6.
The approximated solution is given as:(5.8)u(tn+1)=u2(L)(tn+1).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:(5.9)u1(k+1)=u1(k)−(D(F1(u1(k)))+δ1(u1(k)))−1(F1(u1(k))+ϵu1(k)),where the function δ can be applied as a scalar, for example, δ=10−6,
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,(5.10)F1(u1(tn+1))=u1(tn+1)−u1(tn)−ΔtA(u1(tn+1))u1(tn+1)=0withu1(tn)=cn,F2(u2)=u2(tn+1)−u2(tn)−ΔtB(u2(tn+1))u2(tn+1)=0withu2(tn)=u1(tn+1),
then we obtain the derivations D(F1(u1(tn+1))) and D(F2(u2(tn+1)))(5.11)D(F1(u1(tn+1)))=1−Δt(A(u1(tn+1))+∂A(u1(tn+1))∂u1(tn+1)u1(tn+1)),D(F2(u2))=1−Δt(B(u2(tn+1))+∂B(u2(tn+1))∂u2(tn+1)u2(tn+1)).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:(5.12)dudt=A(u(t))u(t)+B(u(t))u(t),withu(tn)=cn,where A(u),B(u):X→X 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:(5.13)dui(t)dt=A(ui−1(t))ui(t)+B(ui−1(t))ui−1(t),withui(tn)=cn,dui+1(t)dt=A(ui−1(t))ui(t)+B(ui−1(t))ui+1,withui+1(tn)=cn,where the time-step is τ=tn+1−tn.
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 levelt=tn+1 is defined as cn+1=u2m+2(tn+1).
We assume the operators A(ui−1),B(ui−1):X→X 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 thatA(ui−1),B(ui−1) 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 equationA(ui−1)≈A(ui),
and in the second equation B(ui−1)≈B(ui+1).We have(5.14)∥A(ui−1(tn+1))ui(tn+1)−A(un+1)u(tn+1)∥≤ϵ,with sufficient
iterations i={1,3,…,2m+1}.Remark 5.3.
The 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
[4].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
[4, 17]).Newton Method
The function is given as:(5.15)F(c)=∂c∂t−A(c(t))c(t)−B(c(t))c(t)=0,
The iteration can be computed as:(5.16)c(k+1)=c(k)−D(F(c(k)))−1F(c(k)),
where D(F(c)) is the Jacobian matrix and k=0,1,….
and 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:(5.17)F1(ui)=∂tui−A(ui)ui−B(ui−1)ui−1=0,withui(tn)=cn,F2(ui+2)=∂tui+1−A(ui)ui−B(ui+1)ui+1=0,withui+1(tn)=cn,where the time-step is τ=tn+1−tn.
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:(5.18)ui(k+1)=ui(k)−D(F1(ui(k)))−1(∂tui(k)−A(ui(k))ui(k)−B(ui−1(k))ui−1(k)),withD(F1(ui(k)))=−(A(ui(k))+∂A(ui(k))∂ui(k)ui(k)),k=0,1,2,…,K,withui(tn)=cn,ui+1(l+1)=ui+1(l)−D(F2(ui+1(l)))−1(∂tui+1(l)−A(ui(k))ui(k)−B(ui+1(k))ui+1(k))c2(l)),withD(F2(ui+1(l)))=−(B(ui+1(l))+∂B(ui+1(l))∂ui+1(l)ui+1(l)),l=0,1,2,…,L,withui+1(tn)=cn,where the time-step is τ=tn+1−tn.
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 applyui(k+1)=ui(k)−(D(F1(ui(k)))+δ1(ui(k)))−1(F1(ui(k))+ϵui(k)),
and analogously ui+1(l+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.


## 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 form(5.1)dcdt=A(c(t))c(t)+B(c(t))c(t),withc(tn)=cn,where A(c),B(c):X→X 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/dt−A(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=10−4.We assume the spatial discretization, with spatial
grid points,i=1,…,m and obtain the differential equation
system:(5.2)F(c)=(F(c1)F(c2)⋮F(cm)),where c=(c1,…,cm)T and m is the number of spatial grid points.The Jacobian matrix for the equation system is given
as:(5.3)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),where c=(c1,…,cm).The modified Jacobian is given as:(5.4)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)),where c=(c1,…,cn).By embedding the sequential splitting method we obtain
the following algorithm.  We decouple
into two equation systems:(5.5)F1(u1)=∂tu1−A(u1)u1=0withu1(tn)=cn,F2(u2)=∂tu2−B(u2)u2=0withu2(tn)=u1(tn+1),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 operatorsA 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:(5.6)u1(k+1)=u1(k)−D(F1(u1(k)))−1(∂tu1(k)−A(u1(k))u1(k)),withD(F1(u1(k)))=∂∂u1(k)(∂tu1(k)−A(u1(k))−∂A(u1(k))∂u1(k)u1(k)),u1(k)(tn)=cn,k=0,1,2,…,K,u2(l+1)=u2(l)−D(F2(u2(l)))−1(∂tu2(l)−B(u2(l))u2(l)),withD(F2(u2(l)))=∂∂u1(k)(∂tu2(k)−B(u2(l))−∂B(u2(l))∂u2(l)u2(l)),u2(l)(tn)=u1K(tn+1),l=0,1,2,…,L.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:
(5.7)|u1(K)(tn+1)−u1(K−1)(tn+1)|≤err,|u2(L)(tn+1)−u2(L−1)(tn+1)|≤err,
where err is the error bound, for example, er r=10−6.
The approximated solution is given as:(5.8)u(tn+1)=u2(L)(tn+1).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:(5.9)u1(k+1)=u1(k)−(D(F1(u1(k)))+δ1(u1(k)))−1(F1(u1(k))+ϵu1(k)),where the function δ can be applied as a scalar, for example, δ=10−6,
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,(5.10)F1(u1(tn+1))=u1(tn+1)−u1(tn)−ΔtA(u1(tn+1))u1(tn+1)=0withu1(tn)=cn,F2(u2)=u2(tn+1)−u2(tn)−ΔtB(u2(tn+1))u2(tn+1)=0withu2(tn)=u1(tn+1),
then we obtain the derivations D(F1(u1(tn+1))) and D(F2(u2(tn+1)))(5.11)D(F1(u1(tn+1)))=1−Δt(A(u1(tn+1))+∂A(u1(tn+1))∂u1(tn+1)u1(tn+1)),D(F2(u2))=1−Δt(B(u2(tn+1))+∂B(u2(tn+1))∂u2(tn+1)u2(tn+1)).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:(5.12)dudt=A(u(t))u(t)+B(u(t))u(t),withu(tn)=cn,where A(u),B(u):X→X 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:(5.13)dui(t)dt=A(ui−1(t))ui(t)+B(ui−1(t))ui−1(t),withui(tn)=cn,dui+1(t)dt=A(ui−1(t))ui(t)+B(ui−1(t))ui+1,withui+1(tn)=cn,where the time-step is τ=tn+1−tn.
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 levelt=tn+1 is defined as cn+1=u2m+2(tn+1).
We assume the operators A(ui−1),B(ui−1):X→X 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 thatA(ui−1),B(ui−1) 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 equationA(ui−1)≈A(ui),
and in the second equation B(ui−1)≈B(ui+1).We have(5.14)∥A(ui−1(tn+1))ui(tn+1)−A(un+1)u(tn+1)∥≤ϵ,with sufficient
iterations i={1,3,…,2m+1}.Remark 5.3.
The 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
[4].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
[4, 17]).Newton Method
The function is given as:(5.15)F(c)=∂c∂t−A(c(t))c(t)−B(c(t))c(t)=0,
The iteration can be computed as:(5.16)c(k+1)=c(k)−D(F(c(k)))−1F(c(k)),
where D(F(c)) is the Jacobian matrix and k=0,1,….
and 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:(5.17)F1(ui)=∂tui−A(ui)ui−B(ui−1)ui−1=0,withui(tn)=cn,F2(ui+2)=∂tui+1−A(ui)ui−B(ui+1)ui+1=0,withui+1(tn)=cn,where the time-step is τ=tn+1−tn.
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:(5.18)ui(k+1)=ui(k)−D(F1(ui(k)))−1(∂tui(k)−A(ui(k))ui(k)−B(ui−1(k))ui−1(k)),withD(F1(ui(k)))=−(A(ui(k))+∂A(ui(k))∂ui(k)ui(k)),k=0,1,2,…,K,withui(tn)=cn,ui+1(l+1)=ui+1(l)−D(F2(ui+1(l)))−1(∂tui+1(l)−A(ui(k))ui(k)−B(ui+1(k))ui+1(k))c2(l)),withD(F2(ui+1(l)))=−(B(ui+1(l))+∂B(ui+1(l))∂ui+1(l)ui+1(l)),l=0,1,2,…,L,withui+1(tn)=cn,where the time-step is τ=tn+1−tn.
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 applyui(k+1)=ui(k)−(D(F1(ui(k)))+δ1(ui(k)))−1(F1(ui(k))+ϵui(k)),
and analogously ui+1(l+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:(6.1)∂u(t)∂t=(λ1+λ3)u(t)+(λ2+λ4)(u(t))p,t∈[0,T],withu(0)=1,where the analytical solution
can be derived as (see also [16]): (6.2)u(t)=exp((λ1+λ3)t)[−λ2+λ4λ1+λ3exp((λ1+λ3)(p−1)t)+c]1/(1−p).Usingu(0)=1 we find that c=1+(λ2+λ4)/(λ1+λ3),
so (6.3)u(t)=exp((λ1+λ3)t){1+λ2+λ4λ1+λ3[1−exp((λ1+λ3)(p−1)t)]}1/(1−p).We choosep=2, λ1=−1, λ2=−0.5, λ3=−100, λ4=−20 and, for example, Δt=10−2.The analytical solutions can be given
as:(6.4)u(t)1−p=u0exp((1−p)(λ1+λ3)t)+λ2+λ4λ1+λ3(exp((1−p)(λ1+λ3)t)−1).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:(6.5)du1(t)dt=A(u1(t))u1(t),withu1(tn)=un,du2(t)dt=B(u2(t))u2,withu2(tn)=u1n+1,  with the nonlinear operators A(u)u=λ1u(t)+λ2(u(t))p−1u,B(u)u=λ3u(t)+λ4(u(t))p−1u.
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:(6.6)u1(t)1−p=u(tn)exp((1−p)(λ1)t)+λ2λ1(exp((1−p)(λ1)t)−1),u2(tn+1)1−p=u1(tn+1)exp((1−p)(λ3)t)+λ4λ3(exp((1−p)(λ3)t)−1), 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:(6.7)du1(t)dt=A(u1(t))u1(t),withu1(tn)=un,du2(t)dt=B(u2(t))u2,withu2(tn)=u1n+1, with the nonlinear operators A(u)u=λ1u(t)+λ2(u(t))p−1u,B(u)u=λ3u(t)+λ4(u(t))p−1u.
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 as(6.8)u1(k+1)=u1(k)−D(F1(u1(k)))−1(∂tu1(k)−A(u1(k))u1(k)),withD(F1(u1(k)))=−(A(u1(k))+∂A(u1(k))∂u1(k)u1(k)),u1(k)(tn)=cn,k=0,1,2,…,K,u2(l+1)=u2(l)−D(F2(u2(l)))−1(∂tu2(l)−B(u2(l))u2(l)),withD(F2(u2(l)))=−(B(u2(l))+∂B(u2(l))∂u2(l)u2(l)),u2(l)(tn)=u1K(tn+1),l=0,1,2,…,L, where we discretize the equations and obtain the
discretized operators:(6.9)∂tu1(k)−A(u1(k))u1(k)=0,as(6.10)F1(u1(tn+1))u1(k)(tn+1)−u1(tn)−ΔtA(u1(k)(tn+1))u1(k)(tn+1)=0,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(6.11)∂tu2(l)−B(u2(l))u2(l),as(6.12)F2(u2(tn+1))=u2(l)(tn+1)−u2(tn)−ΔtB(u2(l)(tn+1))u2(l)(tn+1)=0,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:(6.13)D(F1(u1(tn+1)))=1−Δt(A(u1(tn+1))+∂A(u1(tn+1))∂u1(tn+1)u1(tn+1)),D(F2(u2))=1−Δt(B(u2(tn+1))+∂B(u2(tn+1))∂u2(tn+1)u2(tn+1)).(3) The standard iterative operator-splitting method
is given as follows.We apply the quasilinear iterative operator-splitting
method:(6.14)dui(t)dt=A(ui−1(t))ui(t)+B(ui−1(t))ui−1(t),withui(tn)=un,dui+1(t)dt=A(ui−1(t))ui(t)+B(ui−1(t))ui+1,withui+1(tn)=un, with the nonlinear operators A(u)u=λ1u(t)+λ2(u(t))p−1u,B(u)u=λ3u(t)+λ4(u(t))p−1u. 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:(6.15)ui(t)=unexp(A(ui−1(t))t)+(A(ui−1(t)))−1(B(ui−1(t))ui−1(t))(1−exp(A(ui−1(t))t)),ui+1(t)=unexp(B(ui−1(t))t)+(B(ui−1(t)))−1(A(ui−1(t))ui−1(t))(1−exp(B(ui−1(t))t)).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:(6.16)dui(t)dt=A(ui(t))ui(t)+B(ui−1(t))ui−1(t),withui(tn)=un,dui+1(t)dt=A(ui(t))ui(t)+B(ui+1(t))ui+1,withui+1(tn)=un,with the nonlinear operators A(u)u=λ1u(t)+λ2(u(t))p−1u,B(u)u=λ3u(t)+λ4(u(t))p−1u.
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:(6.17)ui(k+1)=ui(k)−D(F1(ui(k)))−1(∂tui(k)−A(ui(k))ui(k)−B(ui−1)ui−1),D(F1(ui(k)))=−(A(ui(k))+∂A(ui(k))∂ui(k)ui(k)),k=0,1,2,…,K,ui(tn)=cn,ui(tn+1)=ui(tn+1)K+1,where|ui(tn+1)K+1−ui(tn+1)K|≤err,ui+1(l+1)=ui+1(l)−D(F2(ui+1(l)))−1(∂tui+1(l)−A(ui)ui−B(ui+1(l))ui+1(l))c2(l)),D(F2(ui+1(l)))=−(B(ui+1(l))+∂B(ui+1(l))∂ui+1(l)ui+1(l)),l=0,1,2,…,L,ui+1(tn)=cn,ui+1(tn+1)=ui+1(tn+1)L+1,where|ui+1(tn+1)L+1−ui+1(tn+1)L|≤err.Here the time-step is τ=tn+1−tn.
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(6.18)∂tui(k)−A(ui(k))ui(k)−B(ui−1)ui−1=0,as(6.19)F1(ui(k)(tn+1))=ui(k)(tn+1)−ui(tn)−Δt(A(ui(k)(tn+1))ui(k)(tn+1)+B(ui−1(tn+1))ui−1(tn+1)),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:(6.20)∂tu2(l)−A(ui)ui−B(u2(l))u2(l)=0,as(6.21)F2(ui+1(l)(tn+1))=ui+1(l)(tn+1)−ui+1(tn)−Δt(A(ui(tn+1))ui(tn+1)+B(ui+1(l)(tn+1))ui+1(l)(tn+1)),where we have the initialization
of the Newton's method as ui+1(0)(tn+1)=0 or ui+1(0)(tn+1)=u(tn).The derivations are given as:(6.22)D(F1(ui(k)(tn+1)))=1−Δt(A(ui+1(k)(tn+1))+∂A(ui+1(k)(tn+1))∂ui+1(k)(tn+1)ui+1(k)(tn+1)),D(F2(ui+1(l)(tn+1)))=1−Δt(B(ui+1(l)(tn+1))+∂B(ui+1(l)(tn+1))∂ui+1(l)(tn+1)ui+1(l)(tn+1)).Our numerical results for the different methods are
presented in Tables1, 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.Table 1
Numerical results for the Bernoulli equation with sequential operator-splitting method.
Time partitionsApproximated solutionError10.00000000006.620107e-04420.00000000005.874983e-02330.00000000006.351951e-01640.00000000001.917794e-01250.00000000022.232302e-010100.00000236262.362646e-006500.00158222871.582229e-003Table 2
Numerical results for the Bernoulli equation with sequential operator-splitting method with embedded Newton's method.
Time partitionsApproximated solutionError10.47051294434.705129e-00140.05469224835.469225e-00250.02699544192.699544e-002100.00080347138.034713e-004150.00000000001.137634e-0441000.00000000001.137634e-044Table 3
Numerical results for the Bernoulli equation with iterative operator-splitting method.
Time partitionsNumber of iter.Approximated solutionError120.01250000001.250000e-002140.29278148102.927815e-0011100.01096671581.096672e-0021500.01095567321.095567e-002520.01099131091.099131e-002540.31528269003.152827e-0015100.01085117231.085117e-0025500.01085096431.085096e-0021020.01089954831.089955e-0021040.24377418562.437742e-00110100.01084263281.084263e-00210500.01084261581.084262e-0025020.01496678821.496679e-0025040.01669139711.669140e-00250100.01574641111.574641e-00250500.01599338641.599339e-00210020.01545722231.545722e-00210040.01600480711.600481e-002100100.01584817811.584818e-002100500.01586731791.586732e-002Table 4
Numerical results for the Bernoulli equation
with iterative operator-splitting method with embedded Newton's method.
Time partitionsNumber of iter.Approximated solutionError120.00000000001.137634e-044140.00000000001.137634e-0441100.00000000001.137634e-0441200.00000000001.137634e-044220.00000000001.137634e-044240.00000000001.137634e-0442100.00000000001.137634e-0442200.00000000001.137634e-044Figure 1
Analytical and approximated solution
with sequential operator-splitting method.Figure 2
Analytical and approximated solution
with sequential operator-splitting method with embedded Newton's method.Figure 3
Analytical and approximated solution
with iterative operator-splitting method.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 to10−3,
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:(6.23)∂tu=−12u∂xu−12u∂yu−12∂xu−12∂yu+μ(∂xxu+∂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],where Ω=[0,1]×[0,1], T=1.25,
and μ is the viscosity.The analytical solution is given as(6.24)uana(x,y,t)=(1+exp(x+y−t2μ))−1+exp(x+y−t2μ),where we compute f(x,y,t) accordingly.We split the convection-diffusion  and the Burgers
equation. The operators are given as:(6.25)A(u)u=−12u∂xu−12u∂yu+12μ(∂xxu+∂yyu),
hence
(6.26)A(u)=12(−u∂x−u∂y+μ(∂xx+∂yy))(theBurgersterm),Bu=−12∂xu−12∂yu+12μ(∂xxu+∂yyu)+f(x,y,t)(theconvection-diffusionterm).For the first
equation we apply the nonlinear Algorithm5.1 and obtain
(6.27)A(ui−1)ui=−12ui−1∂xui−12ui−1∂yui+12μ(∂xxui+∂yyui),Bui−1=12(−∂x−∂y+μ(∂xx+∂yy))ui−1,and we obtain
linear operators, because ui−1 is known from the previous time-step.In the second
equation we obtain by using Algorithm5.1:
(6.28)A(ui−1)ui=−12ui−1∂xui−12ui−1∂yui+12μ(∂xxui+∂yyui),Bui+1=12(−∂x−∂y+μ(∂xx+∂yy))ui+1,and we have
linear operators.We deal with different viscositiesμ as well as different step-sizes in time and
space. We have the following results (see Tables 5 and 6).Table 5
Numerical
results for the mixed convection-diffusion and Burgers equation with viscositμ=0.5,
initial conditiou0(t)=cn,
and four iterations per time-step.
Δx=ΔyΔterrL1errmaxρL1ρmax1/51/200.01370.03541/101/200.00550.01391.32641.34991/201/200.00170.00431.68681.69001/401/208.8839⋅10−53.8893⋅10−44.25883.46631/51/400.01460.03771/101/400.00640.01601.19841.23151/201/400.00260.00631.30041.33751/401/408.2653⋅10−40.00211.64781.6236Table 6
Numerical
results for the mixed convection-diffusion and Burgers equation with viscositμ=5,
initial conditiou0(t)=cn,
and two iterations per time-step.
Δx=ΔyΔterrL1errmaxρL1ρmax1/51/201.3166⋅10−52.9819⋅10−51/101/205.6944⋅10−61.3541⋅10−51.20921.13891/201/201.6986⋅10−64.5816⋅10−61.74521.56341/401/207.8145⋅10−72.0413⋅10−61.12011.16631/51/401.4425⋅10−53.2036⋅10−51/101/407.2343⋅10−61.5762⋅10−50.99571.02331/201/403.0776⋅10−66.7999⋅10−61.23301.21291/401/409.8650⋅10−72.3352⋅10−61.64141.5420Figure4 presents the profile of the 2D linear and nonlinear
convection-diffusion equation.Mixed
convection-diffusion and Burgers equation at initial timet=0.0 (a) and end time t=1.25 (b) for viscosity μ=0.5.
(a)(b)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.(6.29)∂tu=−u⋅∇u+2μ∇(D(u)+13∇u)+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],(enclosedflow),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 functionD(u)=u⋅u+v⋅u 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:(6.30)u1(x,y,t)=(1+exp(x+y−t2μ))−1+exp(x+y−t2μ),u2(x,y,t)=(1+exp(x+y−t2μ))−1+exp(x+y−t2μ). For the
splitting method our operators are given as:
(6.31)A(u)u=−u∇u+2μ∇D(u)(thenonlinearoperator),Bu=23μΔu(thelinearoperator).We deal first with the one-dimensional
case,(6.32)∂tu=−u⋅∂xu+2μ∂x(D(u)+13∂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],(enclosedflow),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:(6.33)A(u)u=−u∂xu+2μ∂xD(u)(thenonlinearoperator),Bu=23μ∂xxu(thelinearoperator).
For the
iterative operator-splitting as fixed point  scheme, we have the following
results (see Tables 7 and 9).Table 7
Numerical
results for the 1D momentum equation witμ=5v=0.001,
initial conditiou0(t)=cn,
and two iterations per time-step.
ΔxΔterrL1errmaxρL1ρmax1/101/200.02130.04951/201/200.02030.04700.06890.07461/401/200.01980.04570.04010.04021/801/200.01950.04500.02160.02091/101/400.01340.03121/201/400.01170.02710.19570.20091/401/400.01080.02490.12130.12111/801/400.01030.02380.06820.06741/101/800.00940.02171/201/800.00730.01690.35910.36411/401/800.00620.01430.24510.24481/801/800.00560.01290.14780.1469Figure5
presents the profile of the 1D momentum equation.1D momentum
equation at initial timet=0.0 (a) and end time t=1.25 (b) for μ=5 and v=0.001.
(a)(b)We have the
following results for the 2D case (see Tables10, 11, and 12).Figure6
presents the profile of the 2D momentum equation.2D momentum
equation at initial timet=0.0 (a), (c)) and end time t=1.25 (b), (d)) for μ=0.5 and v=(1,1)t for the first and second component of the
numerical solution.
(a)(b)(c)(d)For the Newton
operator-splitting method we obtain the following functional matrices for the
one-dimensional case:(6.34)DF(u)=(4μ−1)∂xu,D(F(u))=−(∂u1F1(u)∂u2F1(u)∂u1F2(u)∂u2F2(u))=−(−∂xu1+4μ∂xu1−∂xu2+4μ∂xu2−∂yu1+4μ∂yu1−∂yu2+4μ∂yu2)=(4μ−1)∇u.For the two-dimensional case, we
use:(6.35)A(u)u=−u∇u+2μ∇D(u)=−(u1∂xu1+u2∂xu2u1∂yu1+u2∂yu2)+2μ(2u1∂xu1+2u2∂xu2+v1∂xu1+v2∂xu22u1∂yu1+2u2∂yu2+v1∂yu1+v2∂yu2).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.Table 8
Numerical
results for the 1D momentum equation witμ=5v=0.001,
initial conditiou0(t)=cn,
two iterations per time-step anK=1 using Newton iterative method.
ΔxΔterrL1errmaxρL1ρmax1/101/200.01800.04351/201/200.01200.02760.58670.65501/401/200.00950.02270.33110.28701/801/200.00850.02080.17060.12311/101/400.01720.04591/201/400.01250.03050.46520.58841/401/400.01080.02530.23660.26981/801/400.00970.02350.11910.11111/101/800.01660.04751/201/800.01320.03380.33270.49171/401/800.01190.02800.16400.27341/801/800.01120.02650.08020.0779Table 9
Numerical
results for the 1D momentum equation witμ=50v=0.1,
initial conditiou0(t)=cn,
and two iterations per time-step.
ΔxΔterrL1errmaxρL1ρmax1/101/202.7352⋅10−66.4129⋅10−61/201/202.3320⋅10−65.4284⋅10−60.23010.24041/401/202.1144⋅10−64.9247⋅10−60.14130.14051/801/202.0021⋅10−64.6614⋅10−60.07870.07931/101/402.1711⋅10−65.2875⋅10−61/201/401.7001⋅10−64.1292⋅10−60.35280.35671/401/401.4388⋅10−63.4979⋅10−60.24080.23941/801/401.3023⋅10−63.1694⋅10−60.14380.14231/101/801.6788⋅10−64.1163⋅10−61/201/801.1870⋅10−62.9138⋅10−60.50010.49841/401/809.1123⋅10−72.2535⋅10−60.38140.37071/801/807.6585⋅10−71.9025⋅10−60.25070.2443Table 10
Numerical
results for the 2D momentum equation withμ=2,v=(1,1)t,
initial conditionu0(t)=cn, and two
iterations per time-step.
Δx=ΔyΔterrL1errmaxρL1ρmaxerrL1errmaxρL1ρmax1st c.1st c.1st c.1st c.2nd c.2nd c.2nd c.2nd c.1/51/200.00270.01120.01450.03211/101/200.00160.00390.74251.52300.00330.00722.15262.15191/201/200.00070.00221.27120.85970.00210.00420.63910.79671/51/400.00450.01480.02880.06011/101/400.00320.00880.51240.74970.01250.02391.20121.33411/201/400.00140.00341.16931.37640.00290.00542.12632.13251/51/800.01360.04250.04930.11111/101/800.00800.02410.76790.81970.02780.05720.82850.95791/201/800.00390.01131.01661.08720.01150.02311.27461.3058Table 11
Numerical
results for the 2D momentum equation for the first component withμ=50,v=(100,0.01)t,
initial conditionu0(t)=cn,
and two iterations per time-step.
Δx=ΔyΔterrL1errmaxρL1ρmax1st c.1st c.1st c.1st c.1/51/201.5438⋅10−53.4309⋅10−51/101/204.9141⋅10−61.0522⋅10−51.65151.70521/201/201.5506⋅10−62.9160⋅10−61.66411.85131/51/402.8839⋅10−55.5444⋅10−51/101/401.3790⋅10−52.3806⋅10−51.06451.21971/201/403.8495⋅10−66.8075⋅10−61.84081.80611/51/803.1295⋅10−55.5073⋅10−51/101/801.7722⋅10−52.6822⋅10−50.82041.03791/201/807.6640⋅10−61.1356⋅10−51.20941.2400Table 12
Numerical
results for the 2D momentum equation for the second component withμ=50,v=(100,0.01)t,
initial conditionu0(t)=cn,
and two iterations per time-step.
Δx=ΔyΔterrL1errmaxρL1ρmax2nd c.2nd c.2nd c.2nd c.1/51/204.3543⋅10−51.4944⋅10−41/101/203.3673⋅10−57.9483⋅10−50.37080.91091/201/202.6026⋅10−55.8697⋅10−50.37170.43741/51/403.4961⋅10−52.2384⋅10−41/101/401.7944⋅10−58.9509⋅10−50.96221.32241/201/401.5956⋅10−53.6902⋅10−50.16951.27831/51/809.9887⋅10−53.3905⋅10−41/101/803.5572⋅10−51.3625⋅10−41.48961.31531/201/801.0557⋅10−54.4096⋅10−51.75251.6275Remark 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, largerμ 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.


## 6.1. First Numerical Example: Bernoulli Equation

As a nonlinear
differential example, we choose the Bernoulli equation:(6.1)∂u(t)∂t=(λ1+λ3)u(t)+(λ2+λ4)(u(t))p,t∈[0,T],withu(0)=1,where the analytical solution
can be derived as (see also [16]): (6.2)u(t)=exp((λ1+λ3)t)[−λ2+λ4λ1+λ3exp((λ1+λ3)(p−1)t)+c]1/(1−p).Usingu(0)=1 we find that c=1+(λ2+λ4)/(λ1+λ3),
so (6.3)u(t)=exp((λ1+λ3)t){1+λ2+λ4λ1+λ3[1−exp((λ1+λ3)(p−1)t)]}1/(1−p).We choosep=2, λ1=−1, λ2=−0.5, λ3=−100, λ4=−20 and, for example, Δt=10−2.The analytical solutions can be given
as:(6.4)u(t)1−p=u0exp((1−p)(λ1+λ3)t)+λ2+λ4λ1+λ3(exp((1−p)(λ1+λ3)t)−1).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:(6.5)du1(t)dt=A(u1(t))u1(t),withu1(tn)=un,du2(t)dt=B(u2(t))u2,withu2(tn)=u1n+1,  with the nonlinear operators A(u)u=λ1u(t)+λ2(u(t))p−1u,B(u)u=λ3u(t)+λ4(u(t))p−1u.
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:(6.6)u1(t)1−p=u(tn)exp((1−p)(λ1)t)+λ2λ1(exp((1−p)(λ1)t)−1),u2(tn+1)1−p=u1(tn+1)exp((1−p)(λ3)t)+λ4λ3(exp((1−p)(λ3)t)−1), 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:(6.7)du1(t)dt=A(u1(t))u1(t),withu1(tn)=un,du2(t)dt=B(u2(t))u2,withu2(tn)=u1n+1, with the nonlinear operators A(u)u=λ1u(t)+λ2(u(t))p−1u,B(u)u=λ3u(t)+λ4(u(t))p−1u.
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 as(6.8)u1(k+1)=u1(k)−D(F1(u1(k)))−1(∂tu1(k)−A(u1(k))u1(k)),withD(F1(u1(k)))=−(A(u1(k))+∂A(u1(k))∂u1(k)u1(k)),u1(k)(tn)=cn,k=0,1,2,…,K,u2(l+1)=u2(l)−D(F2(u2(l)))−1(∂tu2(l)−B(u2(l))u2(l)),withD(F2(u2(l)))=−(B(u2(l))+∂B(u2(l))∂u2(l)u2(l)),u2(l)(tn)=u1K(tn+1),l=0,1,2,…,L, where we discretize the equations and obtain the
discretized operators:(6.9)∂tu1(k)−A(u1(k))u1(k)=0,as(6.10)F1(u1(tn+1))u1(k)(tn+1)−u1(tn)−ΔtA(u1(k)(tn+1))u1(k)(tn+1)=0,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(6.11)∂tu2(l)−B(u2(l))u2(l),as(6.12)F2(u2(tn+1))=u2(l)(tn+1)−u2(tn)−ΔtB(u2(l)(tn+1))u2(l)(tn+1)=0,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:(6.13)D(F1(u1(tn+1)))=1−Δt(A(u1(tn+1))+∂A(u1(tn+1))∂u1(tn+1)u1(tn+1)),D(F2(u2))=1−Δt(B(u2(tn+1))+∂B(u2(tn+1))∂u2(tn+1)u2(tn+1)).(3) The standard iterative operator-splitting method
is given as follows.We apply the quasilinear iterative operator-splitting
method:(6.14)dui(t)dt=A(ui−1(t))ui(t)+B(ui−1(t))ui−1(t),withui(tn)=un,dui+1(t)dt=A(ui−1(t))ui(t)+B(ui−1(t))ui+1,withui+1(tn)=un, with the nonlinear operators A(u)u=λ1u(t)+λ2(u(t))p−1u,B(u)u=λ3u(t)+λ4(u(t))p−1u. 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:(6.15)ui(t)=unexp(A(ui−1(t))t)+(A(ui−1(t)))−1(B(ui−1(t))ui−1(t))(1−exp(A(ui−1(t))t)),ui+1(t)=unexp(B(ui−1(t))t)+(B(ui−1(t)))−1(A(ui−1(t))ui−1(t))(1−exp(B(ui−1(t))t)).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:(6.16)dui(t)dt=A(ui(t))ui(t)+B(ui−1(t))ui−1(t),withui(tn)=un,dui+1(t)dt=A(ui(t))ui(t)+B(ui+1(t))ui+1,withui+1(tn)=un,with the nonlinear operators A(u)u=λ1u(t)+λ2(u(t))p−1u,B(u)u=λ3u(t)+λ4(u(t))p−1u.
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:(6.17)ui(k+1)=ui(k)−D(F1(ui(k)))−1(∂tui(k)−A(ui(k))ui(k)−B(ui−1)ui−1),D(F1(ui(k)))=−(A(ui(k))+∂A(ui(k))∂ui(k)ui(k)),k=0,1,2,…,K,ui(tn)=cn,ui(tn+1)=ui(tn+1)K+1,where|ui(tn+1)K+1−ui(tn+1)K|≤err,ui+1(l+1)=ui+1(l)−D(F2(ui+1(l)))−1(∂tui+1(l)−A(ui)ui−B(ui+1(l))ui+1(l))c2(l)),D(F2(ui+1(l)))=−(B(ui+1(l))+∂B(ui+1(l))∂ui+1(l)ui+1(l)),l=0,1,2,…,L,ui+1(tn)=cn,ui+1(tn+1)=ui+1(tn+1)L+1,where|ui+1(tn+1)L+1−ui+1(tn+1)L|≤err.Here the time-step is τ=tn+1−tn.
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(6.18)∂tui(k)−A(ui(k))ui(k)−B(ui−1)ui−1=0,as(6.19)F1(ui(k)(tn+1))=ui(k)(tn+1)−ui(tn)−Δt(A(ui(k)(tn+1))ui(k)(tn+1)+B(ui−1(tn+1))ui−1(tn+1)),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:(6.20)∂tu2(l)−A(ui)ui−B(u2(l))u2(l)=0,as(6.21)F2(ui+1(l)(tn+1))=ui+1(l)(tn+1)−ui+1(tn)−Δt(A(ui(tn+1))ui(tn+1)+B(ui+1(l)(tn+1))ui+1(l)(tn+1)),where we have the initialization
of the Newton's method as ui+1(0)(tn+1)=0 or ui+1(0)(tn+1)=u(tn).The derivations are given as:(6.22)D(F1(ui(k)(tn+1)))=1−Δt(A(ui+1(k)(tn+1))+∂A(ui+1(k)(tn+1))∂ui+1(k)(tn+1)ui+1(k)(tn+1)),D(F2(ui+1(l)(tn+1)))=1−Δt(B(ui+1(l)(tn+1))+∂B(ui+1(l)(tn+1))∂ui+1(l)(tn+1)ui+1(l)(tn+1)).Our numerical results for the different methods are
presented in Tables1, 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.Table 1
Numerical results for the Bernoulli equation with sequential operator-splitting method.
Time partitionsApproximated solutionError10.00000000006.620107e-04420.00000000005.874983e-02330.00000000006.351951e-01640.00000000001.917794e-01250.00000000022.232302e-010100.00000236262.362646e-006500.00158222871.582229e-003Table 2
Numerical results for the Bernoulli equation with sequential operator-splitting method with embedded Newton's method.
Time partitionsApproximated solutionError10.47051294434.705129e-00140.05469224835.469225e-00250.02699544192.699544e-002100.00080347138.034713e-004150.00000000001.137634e-0441000.00000000001.137634e-044Table 3
Numerical results for the Bernoulli equation with iterative operator-splitting method.
Time partitionsNumber of iter.Approximated solutionError120.01250000001.250000e-002140.29278148102.927815e-0011100.01096671581.096672e-0021500.01095567321.095567e-002520.01099131091.099131e-002540.31528269003.152827e-0015100.01085117231.085117e-0025500.01085096431.085096e-0021020.01089954831.089955e-0021040.24377418562.437742e-00110100.01084263281.084263e-00210500.01084261581.084262e-0025020.01496678821.496679e-0025040.01669139711.669140e-00250100.01574641111.574641e-00250500.01599338641.599339e-00210020.01545722231.545722e-00210040.01600480711.600481e-002100100.01584817811.584818e-002100500.01586731791.586732e-002Table 4
Numerical results for the Bernoulli equation
with iterative operator-splitting method with embedded Newton's method.
Time partitionsNumber of iter.Approximated solutionError120.00000000001.137634e-044140.00000000001.137634e-0441100.00000000001.137634e-0441200.00000000001.137634e-044220.00000000001.137634e-044240.00000000001.137634e-0442100.00000000001.137634e-0442200.00000000001.137634e-044Figure 1
Analytical and approximated solution
with sequential operator-splitting method.Figure 2
Analytical and approximated solution
with sequential operator-splitting method with embedded Newton's method.Figure 3
Analytical and approximated solution
with iterative operator-splitting method.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 to10−3,
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:(6.23)∂tu=−12u∂xu−12u∂yu−12∂xu−12∂yu+μ(∂xxu+∂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],where Ω=[0,1]×[0,1], T=1.25,
and μ is the viscosity.The analytical solution is given as(6.24)uana(x,y,t)=(1+exp(x+y−t2μ))−1+exp(x+y−t2μ),where we compute f(x,y,t) accordingly.We split the convection-diffusion  and the Burgers
equation. The operators are given as:(6.25)A(u)u=−12u∂xu−12u∂yu+12μ(∂xxu+∂yyu),
hence
(6.26)A(u)=12(−u∂x−u∂y+μ(∂xx+∂yy))(theBurgersterm),Bu=−12∂xu−12∂yu+12μ(∂xxu+∂yyu)+f(x,y,t)(theconvection-diffusionterm).For the first
equation we apply the nonlinear Algorithm5.1 and obtain
(6.27)A(ui−1)ui=−12ui−1∂xui−12ui−1∂yui+12μ(∂xxui+∂yyui),Bui−1=12(−∂x−∂y+μ(∂xx+∂yy))ui−1,and we obtain
linear operators, because ui−1 is known from the previous time-step.In the second
equation we obtain by using Algorithm5.1:
(6.28)A(ui−1)ui=−12ui−1∂xui−12ui−1∂yui+12μ(∂xxui+∂yyui),Bui+1=12(−∂x−∂y+μ(∂xx+∂yy))ui+1,and we have
linear operators.We deal with different viscositiesμ as well as different step-sizes in time and
space. We have the following results (see Tables 5 and 6).Table 5
Numerical
results for the mixed convection-diffusion and Burgers equation with viscositμ=0.5,
initial conditiou0(t)=cn,
and four iterations per time-step.
Δx=ΔyΔterrL1errmaxρL1ρmax1/51/200.01370.03541/101/200.00550.01391.32641.34991/201/200.00170.00431.68681.69001/401/208.8839⋅10−53.8893⋅10−44.25883.46631/51/400.01460.03771/101/400.00640.01601.19841.23151/201/400.00260.00631.30041.33751/401/408.2653⋅10−40.00211.64781.6236Table 6
Numerical
results for the mixed convection-diffusion and Burgers equation with viscositμ=5,
initial conditiou0(t)=cn,
and two iterations per time-step.
Δx=ΔyΔterrL1errmaxρL1ρmax1/51/201.3166⋅10−52.9819⋅10−51/101/205.6944⋅10−61.3541⋅10−51.20921.13891/201/201.6986⋅10−64.5816⋅10−61.74521.56341/401/207.8145⋅10−72.0413⋅10−61.12011.16631/51/401.4425⋅10−53.2036⋅10−51/101/407.2343⋅10−61.5762⋅10−50.99571.02331/201/403.0776⋅10−66.7999⋅10−61.23301.21291/401/409.8650⋅10−72.3352⋅10−61.64141.5420Figure4 presents the profile of the 2D linear and nonlinear
convection-diffusion equation.Mixed
convection-diffusion and Burgers equation at initial timet=0.0 (a) and end time t=1.25 (b) for viscosity μ=0.5.
(a)(b)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.(6.29)∂tu=−u⋅∇u+2μ∇(D(u)+13∇u)+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],(enclosedflow),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 functionD(u)=u⋅u+v⋅u 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:(6.30)u1(x,y,t)=(1+exp(x+y−t2μ))−1+exp(x+y−t2μ),u2(x,y,t)=(1+exp(x+y−t2μ))−1+exp(x+y−t2μ). For the
splitting method our operators are given as:
(6.31)A(u)u=−u∇u+2μ∇D(u)(thenonlinearoperator),Bu=23μΔu(thelinearoperator).We deal first with the one-dimensional
case,(6.32)∂tu=−u⋅∂xu+2μ∂x(D(u)+13∂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],(enclosedflow),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:(6.33)A(u)u=−u∂xu+2μ∂xD(u)(thenonlinearoperator),Bu=23μ∂xxu(thelinearoperator).
For the
iterative operator-splitting as fixed point  scheme, we have the following
results (see Tables 7 and 9).Table 7
Numerical
results for the 1D momentum equation witμ=5v=0.001,
initial conditiou0(t)=cn,
and two iterations per time-step.
ΔxΔterrL1errmaxρL1ρmax1/101/200.02130.04951/201/200.02030.04700.06890.07461/401/200.01980.04570.04010.04021/801/200.01950.04500.02160.02091/101/400.01340.03121/201/400.01170.02710.19570.20091/401/400.01080.02490.12130.12111/801/400.01030.02380.06820.06741/101/800.00940.02171/201/800.00730.01690.35910.36411/401/800.00620.01430.24510.24481/801/800.00560.01290.14780.1469Figure5
presents the profile of the 1D momentum equation.1D momentum
equation at initial timet=0.0 (a) and end time t=1.25 (b) for μ=5 and v=0.001.
(a)(b)We have the
following results for the 2D case (see Tables10, 11, and 12).Figure6
presents the profile of the 2D momentum equation.2D momentum
equation at initial timet=0.0 (a), (c)) and end time t=1.25 (b), (d)) for μ=0.5 and v=(1,1)t for the first and second component of the
numerical solution.
(a)(b)(c)(d)For the Newton
operator-splitting method we obtain the following functional matrices for the
one-dimensional case:(6.34)DF(u)=(4μ−1)∂xu,D(F(u))=−(∂u1F1(u)∂u2F1(u)∂u1F2(u)∂u2F2(u))=−(−∂xu1+4μ∂xu1−∂xu2+4μ∂xu2−∂yu1+4μ∂yu1−∂yu2+4μ∂yu2)=(4μ−1)∇u.For the two-dimensional case, we
use:(6.35)A(u)u=−u∇u+2μ∇D(u)=−(u1∂xu1+u2∂xu2u1∂yu1+u2∂yu2)+2μ(2u1∂xu1+2u2∂xu2+v1∂xu1+v2∂xu22u1∂yu1+2u2∂yu2+v1∂yu1+v2∂yu2).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.Table 8
Numerical
results for the 1D momentum equation witμ=5v=0.001,
initial conditiou0(t)=cn,
two iterations per time-step anK=1 using Newton iterative method.
ΔxΔterrL1errmaxρL1ρmax1/101/200.01800.04351/201/200.01200.02760.58670.65501/401/200.00950.02270.33110.28701/801/200.00850.02080.17060.12311/101/400.01720.04591/201/400.01250.03050.46520.58841/401/400.01080.02530.23660.26981/801/400.00970.02350.11910.11111/101/800.01660.04751/201/800.01320.03380.33270.49171/401/800.01190.02800.16400.27341/801/800.01120.02650.08020.0779Table 9
Numerical
results for the 1D momentum equation witμ=50v=0.1,
initial conditiou0(t)=cn,
and two iterations per time-step.
ΔxΔterrL1errmaxρL1ρmax1/101/202.7352⋅10−66.4129⋅10−61/201/202.3320⋅10−65.4284⋅10−60.23010.24041/401/202.1144⋅10−64.9247⋅10−60.14130.14051/801/202.0021⋅10−64.6614⋅10−60.07870.07931/101/402.1711⋅10−65.2875⋅10−61/201/401.7001⋅10−64.1292⋅10−60.35280.35671/401/401.4388⋅10−63.4979⋅10−60.24080.23941/801/401.3023⋅10−63.1694⋅10−60.14380.14231/101/801.6788⋅10−64.1163⋅10−61/201/801.1870⋅10−62.9138⋅10−60.50010.49841/401/809.1123⋅10−72.2535⋅10−60.38140.37071/801/807.6585⋅10−71.9025⋅10−60.25070.2443Table 10
Numerical
results for the 2D momentum equation withμ=2,v=(1,1)t,
initial conditionu0(t)=cn, and two
iterations per time-step.
Δx=ΔyΔterrL1errmaxρL1ρmaxerrL1errmaxρL1ρmax1st c.1st c.1st c.1st c.2nd c.2nd c.2nd c.2nd c.1/51/200.00270.01120.01450.03211/101/200.00160.00390.74251.52300.00330.00722.15262.15191/201/200.00070.00221.27120.85970.00210.00420.63910.79671/51/400.00450.01480.02880.06011/101/400.00320.00880.51240.74970.01250.02391.20121.33411/201/400.00140.00341.16931.37640.00290.00542.12632.13251/51/800.01360.04250.04930.11111/101/800.00800.02410.76790.81970.02780.05720.82850.95791/201/800.00390.01131.01661.08720.01150.02311.27461.3058Table 11
Numerical
results for the 2D momentum equation for the first component withμ=50,v=(100,0.01)t,
initial conditionu0(t)=cn,
and two iterations per time-step.
Δx=ΔyΔterrL1errmaxρL1ρmax1st c.1st c.1st c.1st c.1/51/201.5438⋅10−53.4309⋅10−51/101/204.9141⋅10−61.0522⋅10−51.65151.70521/201/201.5506⋅10−62.9160⋅10−61.66411.85131/51/402.8839⋅10−55.5444⋅10−51/101/401.3790⋅10−52.3806⋅10−51.06451.21971/201/403.8495⋅10−66.8075⋅10−61.84081.80611/51/803.1295⋅10−55.5073⋅10−51/101/801.7722⋅10−52.6822⋅10−50.82041.03791/201/807.6640⋅10−61.1356⋅10−51.20941.2400Table 12
Numerical
results for the 2D momentum equation for the second component withμ=50,v=(100,0.01)t,
initial conditionu0(t)=cn,
and two iterations per time-step.
Δx=ΔyΔterrL1errmaxρL1ρmax2nd c.2nd c.2nd c.2nd c.1/51/204.3543⋅10−51.4944⋅10−41/101/203.3673⋅10−57.9483⋅10−50.37080.91091/201/202.6026⋅10−55.8697⋅10−50.37170.43741/51/403.4961⋅10−52.2384⋅10−41/101/401.7944⋅10−58.9509⋅10−50.96221.32241/201/401.5956⋅10−53.6902⋅10−50.16951.27831/51/809.9887⋅10−53.3905⋅10−41/101/803.5572⋅10−51.3625⋅10−41.48961.31531/201/801.0557⋅10−54.4096⋅10−51.75251.6275Remark 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, largerμ 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.



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*Source: 307298-2009-02-26.xml*