| Abstract: |
| The Summation-by-Parts (SBP) method is a spatial discretization technique for partial differential equations (PDEs) that employs the SBP operator to approximate spatial partial derivatives. Moreover, some SBP operators provide highly accurate approximations of spatial partial derivatives, and their use is expected to yield high-precision spatial discretization of PDEs. On the other hand, there is a structure-preserving numerical method called the discrete variational derivative (DVD) method by Furihata and Matsuo (2010). The scheme derived from this method is typically second-order accurate in space. More recently, Umezu et al. (2025) have designed a structure-preserving scheme with high-order spatial accuracy based on the DVD method for the Cahn--Hilliard equation with homogeneous Neumann boundary conditions, using the SBP operator and an appropriate projection matrix. However, constructing such a projection matrix is difficult for complex boundary conditions, such as dynamic ones that involve the time derivative of an unknown function. Thus, in this study, instead of using a projection matrix, we incorporated a correction term, the simultaneous approximation (SA) term, corresponding to the residual of the boundary conditions. This enabled us to design a spatially high-accuracy structure-preserving scheme based on the DVD method for PDEs under dynamic boundary conditions. |
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