A deferred correction method is utilized to increase the order of spatial accuracy of the Crank-Nicolson scheme for the numerical solution of the one-dimensional heat equation. The fourth-order methods proposed are the easier development and can be solved by using Thomas algorithms. The stability analysis and numerical experiments have been limited to one-dimensional heat-conducting problems with Dirichlet boundary conditions and initial data.
The desired properties of finite difference schemes are stability, accuracy, and efficiency. These requirements are in conflict with each other. In many applications a high-order accuracy is required in the spatial discretization. To reach better stability, implicit approximation is desired. For a high-order method of traditional type (not a high-order compact (HOC)), the stencil becomes wider with increasing order of accuracy. For a standard centered discretization of order
The development of high-order compact (HOC) schemes [
Another way of preserving a compact stencil at higher time level and reaching high-order spatial accuracy is the deferred correction approach [
In this paper we use the deferred correction technique to obtain fourth-order accurate schemes in space for the one-dimensional heat-conducting problem with Dirichlet boundary conditions. The linear system that needs to be solved at each time step is similar to the standard Crank-Nikolson method of second order which is solved by using Thomas algorithms. The fourth-order deferred (FOD) correction schemes are compared with the fourth-order semi-implicit (FOS) schemes and fourth-order compact (FOC) schemes for the Dirichlet boundary value problems.
A set of schemes are constructed for the one-dimensional heat-conducting problem with Dirichlet boundary conditions and initial data:
The rest of this paper is organized as follows. Section
Let
The application to the well-known Crank-Nikolson scheme to (
A set of fourth-order deferred correction schemes is based on the well-known Crank-Nikolson type of scheme in the following form:
The deferred correction technique [
To preserve a compact three using wide stencil in the finite difference scheme at higher time level
To study the stability of scheme (
Let us recast scheme (
Variation of amplification factor with
Let us first consider the one-dimensional heat conduction problem with initial data and Dirichlet boundary conditions (
Let us briefly represent the main idea and final formulae of compact schemes. Spatial derivatives in the Crank-Nikolson scheme (
In [
In this section, three numerical examples are carried out. The first two are linear heat-conducting problem, with Dirichlet boundary conditions, which are used to confirm our theoretical analysis. Then we apply the FODS to the Burgers equation. For simplicity, we fix our problem domain
The computations are performed using uniform grids of 11, 21, 41, 81, and 161 nodes. The initial and boundary conditions are obtained based on the exact solutions. For the testing purpose only, all computations are performed for
One has
Maximum absolute error, order of convergence, and CPU time in seconds of the FOCs, FODs, and FOSs for test problem (
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One has
Absolute error, the rate of convergence, and CPU time in seconds of the FOCs, FODs, and FOSs for the test problem (
Types of scheme | Grid points | Maximum error | Rate of convergence | Aver. number of iteration | CPU time in sec. |
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FODs |
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The last two columns of Tables
Although the FODs use more computational time as compared with FOCs and FOSs, it is recommended that the construction of FODs can be easily implemented. Moreover, the scheme does not need to store the inverse of coefficient matrices as required in FOCs and FOSs. Therefore, the method is easily extended to multidimensional cases.
It is suggested that the differed correction technique can solve problems which need high accuracy of computational methods. Also this technique can be easily implemented and extended for solving problem with Neumann boundary conditions. In addition, such technique can be easily used to create standard code and applied in case of nonuniform grids.
Considering Burgers equation
This problem was solved using different time step and mesh sizes over the time interval
Maximum absolute error, order of convergence, and CPU time in seconds for Example
Types of scheme | Time step sizes | Maximum error | Rate of convergence | Aver. number of iteration | CPU time in sec. |
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Maximum absolute error, order of convergence, and CPU time in seconds for Example
Types of scheme | Grid points | Maximum error | Rate of convergence | Aver. number of iteration | CPU time in sec. |
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In order to analyze the results found in application to the Burgers equation (
It can be seen from Tables
In this paper, a new set of fourth-order schemes for the one-dimensional heat conduction problem with Dirichlet boundary conditions is constructed using a deferred correction technique. The construction of high-order deferred correction schemes requires only a regular three-point stencil at higher time level which is similar to the standard second-order Crank-Nikolson method. The greatest significance of FODs, compared with FOCs and FOSs, is the easier development and that it can be solved by using Thomas algorithms. Numerical examples confirm the order of accuracy. We also implement our algorithms to nonlinear problems. However, theoretical analysis for nonlinear problems needs further investigation. Posterior idea for this project is to use another way to make
This work is financially supported by the Commission on Higher Education (CHE), the Thailand Research Fund (TRF), the University of Phayao (UP), Project MRG5580014. The author would like to express their deep appreciation to Professor Sergey Meleshko for the kind assistance and valuable advice on the REDUCE calculations.