A granular computing method for nonlinear convection-diffusion equation

This paper introduces a method of solving nonlinear convection-diffusion equation (NCDE), based on the combination of granular computing (GrC) and characteristics finite element method (CFEM). The key idea of the proposed method (denoted as GrC-CFEM) is to reconstruct the solution from coarse-grained layer to fine-grained layer. It first gets the nonlinear solution on the coarse-grained layer, and then the function (Taylor expansion) is applied to linearize the NCDE on the fine-grained layer. Switch to the fine-grained layer, the linear solution is directly derived from the nonlinear solution. The full nonlinear problem is solved only on the coarse-grained layer. Numerical experiments show that the GrC-CFEM can accelerate the convergence and improve the computational efficiency without sacrificing the accuracy.


Introduction
As a key branch of PDEs, the convection-diffusion problem plays an important role in the field of theoretical as well as applied research.
In this paper, we consider the following NCDE in 2 2 : @ @ ^0 is Sobolev Space [1] on : , m is nonnegative integer, the Sobolev Space norm is:   , , , , , , c c a a c c c are positive constant.In most applications, PDEs are often difficult to get the analytic solutions.Therefore, numerical methods must be applied.At present, the major numerical methods are finite element methods(FEM) [2][3][4] [5], finite difference methods (FDM) [6] [7], finite volume methods(FVM) [8][9] [10].However, when the FEM or FDM are used to solve the NCDE, it exhibits excessive numerical diffusion and nonphysical oscillation [11].To address this issue, we use the modified method of CFEM developed by Douglas-Russell [12].But it is of high computational complexity and low efficiency.
Originally, in 1965, Zadeh, proposed the GrC.GrC is a natural mode to simulate human thinking and solve the large-scale problem [13].For many practical problems, we cannot get the exact solution.Instead of searching for the exact solution, one may search for good approximate solutions.The basic components of GrC include granule, granulated views and levels, hierarchies, and granular structures [14].
As far as we know GrC theory is not applied to the NCDE.In this paper, we consider the GrC for NCDE, and focus on the reconstruction of solution from coarsegrained layer to fine-grained layer.We first gets the nonlinear solution on the coarse-grained layer, and then the function ( ) f x is applied to linearizing the NCDE.Switch to the fine-grained layer, the linear solution is directly derived from the nonlinear solution.The full nonlinear problem is solved only on the coarse-grained layer.The new way of GrC-CFEM can accelerate the convergence and improve the computational efficiency without sacrificing the accuracy.The rest of this paper is organized as following.In Section 2, the general theories of Characteristic finite element discretization and Granular computing are described.In Section 3, we illustrate the applications of GrC-CFEM method to the NCDE.Section 4 gives convergence analysis and time complexity analysis.Section 5 presents a numerical experiment.Finally, conclusions are given in Section 6.

Characteristic finite element discretization
The concept of CFEM is now generally well understood, this article only provides a brief introduction.Let Then Eq. (1.1) can be put in form @ @ ^0 , We consider a time step ' t T N and approximate the solution at times Where x x b x c x t is the foot (at level 1 n t ) of the characteristic corresponding to x at the head (at level n t ).We denote the grid by 'h whose grid size is h , h V be a finite-element subspace of 1,f : V W . So, the CFEM for (2.4) is defined: for 1,2, , , The initial approximation 0 h u in h V can be defined as any reasonable approximation of 0 u such as the interpolation of in .h V The (2.6) determines ^ǹ h u uniquely in terms of the data 0 u .

Granular computing
This paper focuses on the reconstruction of solution based on the transformation between granule spaces [16].In figure 1, fine-grained layer and coarse-grained layer are different from granular partitions on the same domain., u x t is NCDE.s and ' s are numerical solutions of the function , u x t on the domain with different granular partitions, respectively.
As we know, it is difficult to directly solve NCDE, furthermore, the acquisition of numerical solution maybe too costly.So, we explore GrC-CFEM to get ' s .The key idea of the proposed method is to reconstruct the solution from coarse-grained layer to fine-grained layer.The full nonlinear problem is solved only on the coarse-grained layer.Comparing with the traditional method, the new method is faster and at a smaller cost.The new way can accelerate the convergence and improve the computational efficiency without sacrificing the accuracy.

Description of the GrC-CFEM
In this section, describe the new method.We combine GrC with CFEM to accelerate the solving process.
The method consists of two major steps: Step1.On the coarse-grained , 'H solve the following nonlinear system for Step2.Switch to the fine-grained , the solution is reconstructed as follows: This linearization is efficient without sacrificing the accuracy associated with the fine-grained solution n

Convergence analysis and time complexity analysis
In this section, the convergence analysis of algorithm are performed.

Convergence analysis
First, we briefly introduce a convergence analysis for the CFEM.Base on it, we give the GrC-CFEM.
We consider (1.1) on : .For H v V the following approximation property holds: where . It is well known that [18], for .
The inverse property on h V holds [19], namely, for And we also have the approximation property: Let u and h u be the respective solutions of (2.4) and (2.6).Under assumptions (a-f) and 't is sufficient small.To derive the following main result, some useful lemmas are needed [20][21].When we have the error estimate: , we have the error estimate: Let u and ˆh u be the respective solutions of (2.4) and (3.2).Under assumptions (a-f), and also assume that 't is sufficient small, 1 t r .We have: .9) for some function u .From (4.10), we have Using arguments similar to [22], plugging (4.12) in to equation (4.11), we can get
Let u and ˆh u be the respective solutions of (2.4) and (3.2).Under assumptions (a-f) and 't is sufficient small, 2 t r , we have:

Time complexity analysis for the CFEM
We will analyze the basic calculation method of CFEM about Eq.(2.6): The pseudo-code for CFEM is as follows: The CFEM Algorithm 01055-p.3The input size of this algorithm is u n m .The key step to solve Eq. (2.6) is to solve nonlinear equation ( ) n h f u .We have following equation by Thayer Series Expansion: The basic operation of solving Thayer Series Expansion is derivation, exponentiation and factorial.Based on input size, the computation times of basic operation is:

Time complexity analysis for the GrC-CFEM
We will analyze the basic calculation method of GrC-CFEM about Eq. (3.1) and (3.CFEM directly solves the nonlinear problem on fine-grained space, but GrC-CFEM can linearize the nonlinear problem.GrC-CFEM involves a nonlinear solution on the coarse-grained space and a linear solution on the fine-grained space.
The input size of this algorithm is u n M .Based on input size, the computation times of basic operation is: During the calculation process of solving Thayer Series Expansion, considering p is limited.So the time complexity of GrC-CFEM is u o n M .So, the GrC-CFEM is faster than the CFEM by one order of magnitude, because there is quadratic relation between m and M .

Numerical experiments
In this section, we provide numerical experiments to demonstrate the efficiency of the GrC-CFEM.
Example1: NCDE with constant coefficients: @ @ ^,     (5. 2) The problem has the following analytical solution with proper initial and boundary conditions: Example2 solved by the GrC-CFEM.We present the results in Figure 5 where  Furthermore.from Figure6, it can be seen that the numerical result also agrees well with the corresponding analytical solution.In addition, a straight comparison between GrC-CFEM and RLBM is presented in Table3.

Conclusions
The key feature of the GrC-CFEM is that it can accelerate the solving process without sacrificing the order of accuracy.Solving the nonlinear problems on the fine-grained layer is reduced to solving a nonlinear system on coarse-grained layer and a linear system on the fine-grained layer.The GrC combined with the CFEM cannot only decreases the numerical oscillation caused by dominating convection, but also saves much computational time for solving the NCDE.

DOI: 10
.1051/ C Owned by the authors, published by EDP Sciences, 201

Figure 1 .
Figure 1.The flow chart of GRC-CFEM can be derived in a similar fashion of the first proof.
m (4.18)During the calculation process of solving Thayer Series Expansion, considering p is limited.So the time complexity of CFEM is u o n m .
2): According to the Eq.(3.1) and (3.2), this algorithm has following pseudo-code: The GrC-CFEM Algorithm Input: // initial-boundary value ' n T t .T is total time, 't is time step Output: ˆh u // Solutions of NCDE 1: for i ← 1 to n do 2: for j ← 0 to M do 3: Figure2 and Figure3 are the solution ˆn h u of (5.1) and its contour-line.The solution ˆn h u obtained by the GrC-CFEM method.The graph shows us how the concentration varies with time and how the concentration varied in space.Figure4is the coarsegrained ( 1 4, H 1 16 h ) and fine-grained ( 1 8, H

Figure 3 .
Figure 3.The contour-line of ˆn h u at different time with

.
In our simulation, the global relative error (GRE) is used to test the accuracy of the model, and defined by t are the numerical and analytical solutions, respectively.
t. we can see the distribution and contourline of numerical solution for example2.

Figure5.
Figure5. the distribution(left) and contour-line(right) of numerical solution for example2

Figure6.
Figure6.The numerical and analytical solution of example2 at different time

Table 2 . 2
[23]e1 shows the2L errors of h u and the CPU time for CFEM over the range of fine-grained 'h .Table2 shows the2L errors of ˆh u and the CPU time for GrC-CFEM when 2 h H . Figure2-4and Table1-2 shows that the solution ˆh u and h u have the same order of accuracy.GrC-CFEM uses less computation time than CFEM.Example2: In this example, we consider twodimensional NCDE with anisotropic diffusion and a source term[23]:

Table 3 .
The distribution and contour-line of numerical solution for example2 (Blank mean that the model is unstable)