Iterative splitting methods have a huge amount to compute matrix exponential. Here, the
acceleration and recovering of higher-order schemes can be achieved. From a theoretical point of
view, iterative splitting methods are at least alternating Picards fix-point iteration schemes.
For practical applications, it is important to compute very fast matrix exponentials. In this
paper, we concentrate on developing fast algorithms to solve the iterative splitting scheme.
First, we reformulate the iterative splitting scheme into an integral notation of matrix exponential.
In this notation, we consider fast approximation schemes to the integral formulations,
also known as

We are motivated to solving multiple phase problems that arose of transport problem in porous media. In the last years, the interest in numerical simulations with multiphase problems, that can be used to model potential damage events has significantly increased in the area of final repositories for chemical or radioactive waste.

Here, the modeling of the underlying porous media, which is the geosphere, spooled with water, is important, and we apply mobile and immobile pores as a two-phase problem in the media. With such a model, we achieve more realistic scenarios of the transported species, see [

The sorption allegorizes the exchange between the solute (mobile) pollutant and at the surface sorbed (immobile) pollutants and appears in the temporal term as well as in the reaction term. The reaction is reversible. The equilibrium-sorption therefore can be declared as a coefficient in the specific terms.

We concentrate on simplified models and taken into account all real conditions for achieving statements of such realistic simulations of potential damage event. Here we have the delicate problem of coupled partial and ordinary differential equations and their underlying multiscale problems. While we deal with splitting methods and decomposing such scale problems, we can overcome such scaling problems, see [

Moreover the computational part is important, while we dealing with large matrices and standard splitting schemes are expensive with respect to compute the exponential matrices. We solve the computational problem with a novel iterative splitting scheme, that concentrate on developing fast algorithms to solve an integral formulation of matrix exponentials. In such a scheme, we consider fast approximation schemes to the integral formulations, also known as

In the following, we describe our model problem. The model equation for the multiple phase equations are given as coupled partial and ordinary differential equations:

effective porosity

concentration of the ^{3}),

concentration of the ^{3}),

concentration of the ^{3}),

concentration of the ^{3}),

velocity through the chamber and porous substrate [

element-specific diffusions-dispersions tensor (cm^{2}/nsec),

decay constant of the

source term of the ^{3}nsec)),

exchange rate between the mobile and immobile concentration (1/nsec),

exchange rate between the mobile and adsorbed concentration or immobile and immobile adsorbed concentration (kinetic controlled sorption) (1/nsec),

stationary electric field in the apparatus (V/cm),

the mobility rate in the electric field, see [^{2}/nsec).

with

The parameters in (

The effective porosity is denoted by

In this paper we concentrate on solving linear evolution equations, such as the differential equation,

Our main focus will be to consider and contrast higher-order algorithms derived from standard schemes as for example Strang-splitting schemes.

We propose iterative splitting schemes as a solver scheme which is simple to implement and parallelisible.

A rewriting to integral formulation, allows to reduce the computation to numerical approximation schemes. While standard schemes has the disadvantage to compute commutators to achieve higher-order schemes, we could speed up our schemes by recursive algorithms. Iterative schemes can be seen as Successive approximations, which are based on recursive integral formulations in which an iterative method is enforce the time dependency.

The paper is outlined as follows. In Section

The following algorithm is based on the iteration with fixed-splitting discretization step-size

In the following we will analyze the convergence and the rate of convergence of the method (

The novelty of the convergence results are the reformulation in integral-notation. Based on this, we can assume to have bounded integral operators which can be estimated and given in a recursive form. Such formulations are known in the work of [

We present the results of the consistency of our iterative method. We assume for the system of operator the generator of a

Let one consider the abstract Cauchy problem in a Hilbert space

Further we assume the estimation of the partial integration of the unbounded operator

Then, we can bound our iterative operator splitting method as

Let us consider the iteration (

For the first iterations, we have

In general, we have:

for the odd iterations,

for the even iterations,

We have the following solutions for the iterative scheme: the solutions for the first two equations are given by the variation of constants:

For the recurrence relations with even and odd iterations, we have the solutions: for the odd iterations:

For the even iterations,

For

We obtain

For

We obtain

For odd and even iterations, the recursive proof is given in the following. For the odd iterations (only iterations over

We obtain

The same idea can be applied to the even iterative scheme and also for alternating

The same idea can be applied to

If we assume the consistency of

The motivation of the splitting method are based on the following observations.

The mobile phase is semidiscretized with fast finite volume methods and can be stored into a stiffness-matrix. We achieve large time steps, if we consider implicit Runge-Kutta methods of lower order (e.g., implicit Euler) as a time discretization method.

The immobile, adsorbed and immobile-adsorbed phases are purely ordinary differential equations and each of them is cheap to solve with explicit Runge-Kutta schemes.

The ODEs can be seen as perturbations and can be solved all explicit in a fast iterative scheme.

For the full equation we consider the following matrix notation:

Further,

Now, we have the following ordinary differential equation:

For such an equation, we apply the decomposition of the matrices:

The equation system is numerically solved by an iterative scheme.

We divide our time interval

We start with

The initial conditions are given with

Compute the fix-point iteration scheme given as

The stop criterion for the time interval

If

If (

The error analysis of the schemes are given in the following Theorem.

Let

The outline of the proof is given in [

In the next section we describe the computation of the integral formulation with

The theoretical ideas can be discussed in the following formulation:

Here, we have to compute the right-hand side as time dependent term, means we evaluate

The

In the following we reduce to a approximation of the fixed right-hand side (means we assume

Later we also follow with more extended schemes.

So the matrix formulation of our scheme is given as

For higher-orders we should also include the full derivations of

Consider the equation

In the following, we discuss the one- and two-side algorithms.

For the implementation of the integral formulation, we have to deal with parallel ideas, which means we select independent parts of the formulation.

Determine the order of the method by fixed iteration number.

Consider the time interval

On each subinterval,

Repeat this procedure for next interval until the desired time

For the one-side scheme, we taken into account of the following commutator relation.

The following relation is given:

The integration is given as

Further, we have the recursive integration

The iterative scheme with the equations

The recursion is given as

For the novel notation, we have embedded the commutator to the computational scheme. For such a scheme we could save to compute additional the commutators.

In the following, we deal with numerical example to verify the theoretical results.

For another example, consider the matrix equation

The Figure

Numerical errors of the standard splitting scheme (a) and the iterative schemes (b) with 1, … , 6 iterative steps.

The Figure

CPU time of the standard splitting scheme (a) and the iterative schemes (b) with 1, … , 6 iterative steps.

We deal in the first with an ODE and separate the complex operator in two simpler operators.

We deal with the

We rewrite (

The operators are splitted in the following way, while operator

The Figure

Numerical errors of the standard splitting scheme (a) and the iterative schemes (b) with 1, … , 6 iterative steps.

The Figure

CPU time of the standard splitting scheme (a) and the iterative schemes (b) with 1, … , 6 iterative steps.

The computational results show the benefit of the iterative schemes. We save computational time and achieve higher-order accuracy. The one-side and two-side schemes have the same results.

We assume to have a large norm of the commutator

In the following, we discuss different splitting ideas.

We split the matrix as

Here we have to deal with highly noncommutative operators and the computational speedup is given in the iterative schemes, while the commutator is not needed to obtain more accurate results, while the standard splitting schemes deal with such commutators for the error reduction.

The Figure

Numerical errors of the standard splitting scheme (a) and the iterative schemes (b) with 1, … , 6 iterative steps.

The Figure

CPU time of the standard splitting scheme (a) and the iterative schemes (b) with 1, … , 6 iterative steps.

Further for the one-side,we obtain more improved results for the following splitting.

In this version, we have

Numerical errors of the standard splitting scheme (a) and the iterative schemes based on one-side to operator

A more delicate problem is given for the stiff matrices.

In this version, we have

Numerical errors of the one-side splitting scheme with

The Figure

CPU time of the one-side splitting scheme with

The iterative schemes with fast computations of the exponential matrices have a speedup. The constant CPU time of the iterative schemes shows that it benefit instead of the expensive standard schemes. Also for stiff problems with multi iterative steps, we reach the same results of the standard

The next example is a simplified real-life problem for a multiphase transport-reaction equation. We deal with mobile and immobile pores in the porous media, such simulations are given for waste scenarios.

We concentrate on the computational benefits of a fast computation of the iterative scheme, given with matrix exponentials.

The equation is given as:

In the following, we deal with the semidiscretized equation given with the matrices:

We have the following two operators for the splitting method

We decouple into the following matrices:

For the operator

Based on the decomposition, operator

The Figure

Numerical errors of the one-side splitting scheme with

For all iterative schemes, we can reach faster results as for the The iterative schemes with fast computations of the exponential matrices standard schemes. With 4-5 iterative steps we obtain more accurate results as we did for the expensive standard schemes. With one-side iterative schemes we reach the best convergence results.

In this work, we have presented a very economical and practical method by successive approximations. Here the idea to decouple the expensive computation of matrix exponential via iterative splitting schemes has the benefit of less computational time. While the error analysis present stable methods and higher-order schemes, the applications show the speedup with the iterative schemes. In, future, we concentrate on matrix dependent scheme, based on iterative splitting algorithms.