Time-dependent parabolic equations are widely used in science and engineering, which are described water head in groundwater modelling, pressure in petroleum reservoir simulation, diffusion phenomena in heat propagation, and atmospheric aerosol transport problems, etc (see [1,2,3, 7,8]). Due to the computational complexities and huge computational costs in applications, the non-overlapping explicit/implicit domain decomposition method have been an important tool for solving parabolic equations [4, 6, 7, 10,11]. Domain decomposition schemes that preserve the mass of the model are important and also required for parallel computations, specially, in long time simulations and for large scale applications. Paper [5] presented an explicit-implicit conservative domain decomposition procedure for parabolic equations, where the fluxes at the sub-domain interfaces were calculated by an average operator from the solutions at the previous time level. Paper [16] studied the cell centered finite difference domain decomposition procedure for the heat equations with constant coefficients in one dimension. Papers [13,14,15] proposed conservative parallel difference schemes for solving 2-dimension (nonlinear) diffusion equation and the theoretical analysis was proved in paper [9]. By the operator splitting technique and the coupling of the solution and its fluxes on staggered meshes, papers [17,18,19] analyzed the new mass-preserving S-DDM scheme for solving parabolic equations and convection-diffusion equations. However, the above conservative domain decomposition methods are only first-order in time. Recently, papers [20,21,22] proposed the time second-order mass-conserved domain decomposition methods for parabolic equations with constant coefficients and variable coefficients, respectively.
In this paper, for improving the accuracy and stability of the mass-conserved schemes in papers [20,21,22], we propose the time second-order and space fourth-order conservative domain decomposition schemes for one-dimension and 2-dimension parabolic equations with Neumann boundary conditions. In the domain decomposition method, we take two steps to solve one-dimension problem. The time extrapolation and local multi-point weighted average schemes are used to approximate the interface fluxes on interfaces of sub-domains, while the interior solutions are computed by the time second-order and space fourth-order implicit schemes in sub-domains. By the operator splitting technique, we take three small time steps (i.e., along
The one dimensional parabolic equations with variable coefficients are considered as follows,
where
Let
For functions
For the stake of simplicity, the domain [0, L] is divided into two sub-domains and
Define
and
Now, we propose the time second-order and space fourth-order conservative domain decomposition scheme of Eqns. (1) in two steps at every time [
The interface fluxes
where
The interior points
where
with the boundary conditions as
and the first time level values
When
Using the boundary condition
Substituting (14) into (13), we have that
This ends the proof of the theorem.
For
Summing with
where
Subtracted (20) from (19), we can obtain that
Similarly, it holds that
Further, we have that
Applying the Taylor format, it holds that
Substituting (23) into (24), we can obtain (16). Similarly, it leads (17).
The two-dimensional parabolic equations are considered as
where Ω = [0, 1] × [0, 1],
For simplicity, we assume that
For simplicity of description, we assume that the domain Ω be divided into 2 × 2 block sub-domains (see Fig. 1). Let
Let
where
Now we describe the algorithm of our time second-order and space fourth-order conserved splitting domain decomposition scheme on Ω1,1 at each time [
Along The intermediate interface fluxes
where
The intermediate variables
Along The interface fluxes
and define
The numerical solutions
Along The intermediate interface fluxes
where
The intermediate variables
The boundary conditions are approximated by
The initial values are computed by
Similar proof as (12), we can obtain the mass along
along
and along
In the section, we present numerical experiments to illustrate the performance of the scheme such as mass conservation, orders of convergence and stability. The domains Ω = [0, 1] × [0, 1] and are divided into 2 × 2 sub-domains. Take uniform mesh steps
and mass errors MassErr = |Mass
Assume that
Errors and ratios of convergence in space for different diffusion 1/10 1/20 1/40 1/80 1E-2 1.0572E-5 6.8466E-7 2.9351E-8 9.9707E-10 ratio - 3.9487 4.5439 4.8796 1E-1 4.6522E-5 1.7552E-6 6.7088E-8 2.9600E-9 ratio - 4.7282 4.7094 4.5024 1 4.2444E-5 1.7928E-6 8.0280E-8 1.5923E-9 ratio - 4.5653 4.4810 5.6559
The time order of convergence of the scheme at time
Errors and ratios of convergence in time for different diffusion 1/1000 1/4000 1/9000 1/16000 1E-2 3.3475E-5 1.6835E-5 1.1157E-5 8.3407E-6 ratio - 1.9695 2.2111 2.3563 1E-1 4.6382E-5 1.7548E-6 2.5642E-7 6.7091E-8 ratio - 2.3621 2.3717 2.3303 1 4.2394E-5 1.7795E-6 2.9232E-7 8.3107E-8 ratio - 2.2872 2.2273 2.1860
From Table 1 and 2, we can see clearly that our scheme are of fourth-order convergence in spatial step and second-order convergence in time step for the cases of different diffusions.
Take the space step
Errors and mass errors for different diffusion 0.01 0.1 0.5 1 1/1000 1.7193E-7 3.1160E-7 2.7226E-7 2.7227E-7 MassErr 2.9837E-18 1.1241E-17 4.0246E-18 4.2986E-17 1/2000 1.7211E-7 3.0987E-7 3.1648E-7 1.4113E-7 MassErr 3.7401E-17 2.6368E-17 2.3835E-17 4.1113E-18 1/3000 1.7218E-7 3.0962E-7 3.2929E-17 1.8791E-7 MassErr 3.1850E-17 4.4409E-17 4.4548E-17 6.9042E-18 1/4000 1.7221-7 3.0956E-7 3.3409E-7 2.0642E-7 MassErr 4.8503-17 1.1796E-18 2.4876E-17 1.7781E-17
The effect of
The effect of 2 3 5 10 20 2 4.2699E+19 3.2818E+02 1.5482E-6 1.6600E-8 2.5564E-7 MassErr 1.4909E+02 6.2177E-16 8.2406E-18 9.7145E-19 2.0761E-17
In Figure 2, we take
In this paper, the time second-order and space fourth-order conserved splitting domain decomposition scheme is developed for solving 2-dimension parabolic equations. In our splitting domain decomposition method, the time extrapolation and local multi-point weighted average schemes are used to approximate the interface fluxes on interfaces of sub-domains, while the interior solutions are computed by the splitting high-order implicit schemes in sub-domains. The analysis of stability and convergence will be studied in further work.