Computational Studies of Reaction-Diffusion Systems by Nonlinear Galerkin Method

This article deals with the computational study of the nonlinear Galerkin method, which is the extension of commonly known Faedo-Galerkin method. The weak formulation of the method is derived and applied to the particular ScottWang-Showalter reaction-diffusion model concerning the problem of combustion of hydrocarbon gases. The proof of convergence of the method based on the method of compactness is introduced. Presented results of numerical simulations are composed of the computational study, where the nonlinear Galerkin method and Faedo-Galerkin method are compared for the problem with analytical solution and the numerical results of the Scott-Wang-Showalter model in 1D.


Introduction
It is well known that many problems often occur when one tries to approximate the complex dynamics of reaction-diffusion equations.Especially the error estimate of common methods grows exponentially in time.One possible approach to overcome this problem, known as the Nonlinear Galerkin method is suggested by Marion and Temam in [1].It is also discussed in [2] and [3].In this paper we discuss this method and its properties, and apply it to the solution of particular reaction-diffusion model and perform a computational study when the method is compared with the commonly known Faedo-Galerkin method.
Consider a system of reaction-diffusion equations where We introduce the space as the Hilbert space with the scalar product and the space as a Hilbert space endowed with the scalar product Let ini H   . Then the weak solution of the problem (1)-( 2) on time interval   0,T is a mapping

 
,T V : 0  such that it satisfies the following equations for each V  : garding the computational time of the Faedo-Galerkin method;  To decrease computational time regarding to the precision of the approximation of the Faedo-Galerkin method.
Analogically to the Faedo-Galerkin method, we search the approximate solution on some finite-dimensional subspace of .Consider a differential equation for the unknown function in the following form: with the initial condition where the mapping is written as   for some linear operator A .The H is a separable Hilbert space with the orthonormal basis composed of eigenvectors of the operator Discretization in the nonlinear Galerkin method is based on the two following steps: 1) Replacing the right hand side by the first order Taylor expansion: where is the Jacobian matrix of .One suggested approach is that the remainder in the Taylor expansion satisfies these following properties (see [1,2,5]): This simplification is implied by a particular nonlinear Galerkin method we are using.Then, the second equation of the system (8) can be written as , we consider the function The equations for the nonlinear Galerkin method can be finally written as the following: .
The degree of approximation is determined by the parameters m and M. We interpret the function as an approximation of solution of (6) for the indices 1, , j m   and 1, , J m    M .We endow these equations with the initial conditions

Application to the Scott-Wang-Showalter Model
We show the application of the nonlinear Galerkin method on the particular reaction-diffusion system.It was experimentally discovered, that there arise patterns created by flames during the combustion of mixed compounds of hydrocarbon gases.This phenomenon is described by the Sal'nikov model (see [4,6,7]), which generates the thermokinetic oscillations.The Sal'nikov's work deals with the problem of the cool flames during the oxidation of hydrocarbon gases.
The scheme of the Sal'nikov's thermokinetic oscillation is the following: In the first reaction, the compound P generates the reactive compound A. In the second reaction, the compound A decomposes to the inert product B during the emergence of heat.The detailed physical point of view is discussed in [6].The system of reaction-diffusion equations for dimensionless concentration  of reaction intermediate A and dimensionless temperature  of reaction compounds is: where the function f is defined as The , , Le   and the  are the parameters of the model,  is the dimensionless time.We complement these equation with the initial conditions ini ini 0 0 , and with the Dirichlet boundary conditions endowed with the homogeneous boundary conditions and with the following initial conditions ini ini 0 0 , exp 1 We consider the unknown functions  and  as mappings from the interval   we introduce the following operator notation for unknown vector Then we define the operator F as Utilizing this notation, we can rewrite the problem ( 16)-(17) as In this case, we consider , the domain and the spaces and . For the application of the nonlinear Galerkin method, we use the orthonormal basis of H composed of eigenvectors of the operator We search the Galerkin approximation of  as the decomposition are given by the following system of differential-algebraic equations: Multiplicating the second equation of ( 22) by   , using simple algebraic manipulations and subtracting it from the first equation of (22), we obtain a linear relation between J  and J  : The coefficients J  are then computed via the relation (23).

Convergence
We prove the convergence of the nonlinear Galerkin method applied to the Scott-Wang-Showalter model.
The most important note is the existence of the invariant region for the Scott-Wang-Showalter model.Its existence was proved in [7].
We introduce the following operator notation where Id is the identical operator.The Jacobian matrix of the operator G is computed as where 1 2 .Hence we can write the following important estimates for the operator : Then the equations for the nonlinear Galerkin method (11) are as follows Problem (25) is the system of differential-algebraic equations solvable on   0, m ue to the theory of ODEs as the algebraic system for m z is uniqu solvable and smoothly depends on m p .The value o m T depends on quality of approximation.
The operator A has the same eigenfunctions as the operator A  : The eigenvalues are 1   . The operator A is (see [8]) positive and self-adjoint.Hence we can define its square root Now we introduce some useful relations between the operator norms which we use in the next part.For more detailed derivation see [4] ,   (29) To prove the convergence of the nonlinear Galerkin method we process the particular sequences in Equation (25).
Using the Young inequality, we obtain According to [1], the expression Using estimates (24), we obtain We use relations ( 27) and (28) on the left hand side of this inequality.Then we obtain the estimate for m z via the Poincaré inequality: A A a Id are linear operators.We suppose that initial condition . Using the Bessel inequality, we obtain following auxiliary estimates We multiply the first equation of (25) scalarly by Using the definition of the square root of operator A, we obtain We use the Young inequality and (24) to estimate the left hand side and then we obtain the auxiliary estimate We integrate the Equation (31) over   0,T : A p T and using(30) we obtain Denoting the the largest one and the the smallest one.
kT Dropping the integral of nonnegative function and using (30) we obtain Using the relations ( 26) and ( 27) we get the inequality We multiply the first equation of (25) scalarly by d d We use the Young inequality for 1   on the last term and estimate the middle term.
Using the boundedness of the operator G (24) we obtain We integrate this inequality over   0,T and use the relations (30): 2 0, ; L T H . 9) Passage to the limit Considering the previous estimates (boundedness of m A p and (27) particularly), we obtain the following properties , ; , 0, ; 0, ; .
Using the Aubi a (see [9]) fo fu , n Lemm r following nction spaces: we obtain that the Banach space , which converges strongly to the limit point p   ak solution rt nd uniqueness of and uniqueness of we In this pa we prove the existence a th Th e weak solution of the Scott-Wang-Showalter model.e existence is proven via the strong convergence of the sequence Multiplying the previous relation by the test function and integrating it over Integrating the left hand side per parts and passing to the limit we obtain Additionally, we consider .Then in sense of distributions.Now, we multiply the Equation (33) by  and integrate it over   0,T .Usi ng integration per parts we obtain Subtracting ( 32) and (34) we get Hence is the weak solution.
To show the uniqueness, we suppose th e are two ons p er different weak soluti  and  , which satisfy We denote      , multiply it by .Then we subtract the previous equations   Finally, using the Young inequality for the last term and Lipschitz condition of operator , we obtain for each 0   , which is the contradiction.

titative
er we deal with the error measurement and nlinear Galerkin method diffusion model.We are the function subspace, where the approximation of the solutions is searched.Before the application on the Scott-Wang-Showalter model, we use the single one-dimensional reaction-diffusion equation with the known analytical solution   , u u t x  as a benchmark for the method.Consider the equation method.The linear systems for correction are solved via Gauss elimination method since they are generally a systems with dense matrices.
We plot the norm of the difference between analytical solution and numerical approximation, i.e.

Simulation 2
Consider equation  

Conclusion
In this paper we applied inear Galerkin method to nonl  the particular system of reaction-diffusion equations in one spatial dimension.As the investigated reaction-diffusion system was chosen the Scott-Wang-Showalter model.We presented the system of differential-algebraic equations for the approximation of the weak solution, proof of existence and uniqueness of the weak solution and the proof of convergence of the nonlinear Galerkin method.We performed quantitative analysis among analytical solution and numerical approximations obtained via the nonlinear Galerkin method and the commonly known Faedo-Galerkin method.It indicates that the nonlinear Galerkin method is more efficient since it conserves the similar level of accuracy with respect to the shorter computational time.

Acknowledgements
Considering the invariant region for the model, the operator G satisfies the Lipschitz condition with the constant for each 0     and each solution with the initial condition inside te invariant region is bounded, i.e.Then we have the following estimates for the right hand sides of the model Since the operator satisfies the Lipschitz condition, w G e can perform the following estimate Copyright © 2013 SciRes.AJCM M. KOLÁŘ 143
the Figures1 and 2.
the Figures 1 and 2. tudies In this section we present the computational results for the Scott-Wang-Showalter model.Cons ering the hodition problem (16)-(17), we ur of the model depending on various initial conditions and various sets of par eters.Additionally, deeper computational study can be found in ime The time evolution of the problem is on the Figure 3.
Partial support of the project No. TA0102871 of th Technological Agency o e Czech Republic, No. outh and Sport of the Czech Republic is Θ e f th SGS11/161/OHK4/3T/14 of the Czech Technical University in Prague, No. MSM 6840770010 of the Ministry of Education, Y acknowledged.
in the space

Table 1 . Computational complexities for testing simula- tions.
proximation from the nonlinear Galerkin method are solved by means of time-adaptive Runge-Kutta-Merson