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[MIS21-P08] Infinite stage CIP method and its relatives
Keywords:CIP method, differential equations, numerical method
A numerical method called the CIP method once dominated (Yabe et al., 2003). The CIP method was originally developed to solve advection equations, but it is applicable to any partial differential equation. The method solves both the original differential equation and an equation of the original equation that is differentiated once in terms of the spatial coordinate. It also takes both the original variable and the spatially differentiated variable of the original variable as dependent variables (field variables). It is characterised by high accuracy due to a large number of available information and a small spatial extent when considered as a grid method (although the setting of boundary conditions is relatively complex). The original CIP method (Tatewaki and Yabe, 1985) is a semi-Lagrangian method, but the original concept has been extended to the finite difference method (the IDO method; Aoki, 1997), the finite element-like method (the CIP-BS method; Utsumi and Kimura, 2004), and the finite volume method (Xiao et al., 2009). Currently, the CIP 'family' deals not only with spatially differentiated quantities, but also with integrated quantities, so they should be called the multi-moment method (Xiao et al., 2009).
The original CIP method considered up to one stage of spatial differentiation, so what would happen if this is extended to multiple stages? The answer is found in the paper on the CIP-BS method (Utsumi and Kimura, 2004). There, it is stated that when the CIP-BS method is applied in infinite steps, interpolation is performed using a piecewise continuous function, which is the expansion of the function using an orthogonal function. However, their paper only implements up to two stages, and the behavior in multi-stage solving methods is unknown. The situation is similar with the finite difference IDO method. In this presentation, we will implement a multi-stage application of the CIP-BS method and the IDO method and discuss its behavior.
The contents of this presentation are as follows. First, the coefficients required for the CIP-BS method and the IDO method are determined in the general form as possible. Let us take as an example the case where the IDO method is applied to a one-dimensional space. The variables at the grid point of interest and those on both sides of it are used, and the dependent variable and its spatial differentials on both sides of the point of interest are subjected to the Taylor expansion with respect to the point of interest. By solving the results of the expansion, approximations for the spatial differential can be obtained. As the number of stages of spatial differentiation increases, the number of coefficients in the Taylor expansion increases, making it difficult to calculate by hand. Therefore, the coefficients are calculated using the mathematical formula processing software Maxima.
Next, some basic differential equations are solved using the determined coefficients. As an example, we will discuss a special case, in which the one-dimensional Poisson equation is solved by the IDO method. If the multi-stage method is applied to the Poisson equation with constant coefficients, the dependent variables are up to the k-th derivative of the original variable. However, the second-order and higher spatial derivatives can be constructed from the original Poisson equation. Therefore, there are two equations to solve: the k-th differentiation of the original equation, and the k+1-th differentiation of the original equation, and the variales to be obtained are only the 0th- and 1st-order differentiation of the dependent variable. In other words, (under the strong constraint that the source terms are known exactly) a solution can be obtained at any accuracy from just two equations. Note that for the Poisson equation with constant coefficients, a method called the Numerov method that can obtain fourth-order accuracy using three-point differences can also be applied. When this is applied to the normal one-stage IDO method, 6th order accuracy can be obtained with 3 points of difference (6 points in terms of information). This Numerov-IDO method can also be applied to the multi-stage IDO, and higher precision differential coefficients can be obtained. (For the Poisson equation, the infinite stage CIP mentioned in the title was not obtained!)
In the presentation, we will apply the IDO method and CIP-BS method to other differential equations. In this case, there is no special property like the Poisson equation, and the solution should be otained using all the k-th differentials of the original equation (k=0,1,...).
The original CIP method considered up to one stage of spatial differentiation, so what would happen if this is extended to multiple stages? The answer is found in the paper on the CIP-BS method (Utsumi and Kimura, 2004). There, it is stated that when the CIP-BS method is applied in infinite steps, interpolation is performed using a piecewise continuous function, which is the expansion of the function using an orthogonal function. However, their paper only implements up to two stages, and the behavior in multi-stage solving methods is unknown. The situation is similar with the finite difference IDO method. In this presentation, we will implement a multi-stage application of the CIP-BS method and the IDO method and discuss its behavior.
The contents of this presentation are as follows. First, the coefficients required for the CIP-BS method and the IDO method are determined in the general form as possible. Let us take as an example the case where the IDO method is applied to a one-dimensional space. The variables at the grid point of interest and those on both sides of it are used, and the dependent variable and its spatial differentials on both sides of the point of interest are subjected to the Taylor expansion with respect to the point of interest. By solving the results of the expansion, approximations for the spatial differential can be obtained. As the number of stages of spatial differentiation increases, the number of coefficients in the Taylor expansion increases, making it difficult to calculate by hand. Therefore, the coefficients are calculated using the mathematical formula processing software Maxima.
Next, some basic differential equations are solved using the determined coefficients. As an example, we will discuss a special case, in which the one-dimensional Poisson equation is solved by the IDO method. If the multi-stage method is applied to the Poisson equation with constant coefficients, the dependent variables are up to the k-th derivative of the original variable. However, the second-order and higher spatial derivatives can be constructed from the original Poisson equation. Therefore, there are two equations to solve: the k-th differentiation of the original equation, and the k+1-th differentiation of the original equation, and the variales to be obtained are only the 0th- and 1st-order differentiation of the dependent variable. In other words, (under the strong constraint that the source terms are known exactly) a solution can be obtained at any accuracy from just two equations. Note that for the Poisson equation with constant coefficients, a method called the Numerov method that can obtain fourth-order accuracy using three-point differences can also be applied. When this is applied to the normal one-stage IDO method, 6th order accuracy can be obtained with 3 points of difference (6 points in terms of information). This Numerov-IDO method can also be applied to the multi-stage IDO, and higher precision differential coefficients can be obtained. (For the Poisson equation, the infinite stage CIP mentioned in the title was not obtained!)
In the presentation, we will apply the IDO method and CIP-BS method to other differential equations. In this case, there is no special property like the Poisson equation, and the solution should be otained using all the k-th differentials of the original equation (k=0,1,...).