Transformed-space nonuniform pseudospectral time-domain algorithm

Analysis of photonic band‐gap structures in stratified medium

“…As stated in Leung and Chen (2001), it should be noticed that the derivative ›u/›x depends solely on the spatial distribution. Thus, it is necessary to be computed only once, i.e.…”

“…However, the use of FT gives rise to a constraint that the spatial distribution must be kept uniform. Therefore, in order to make the PSTD be flexible as in non-uniform FDTD (Choi and Hoefer, 1986), a nu-PSTD approach is proposed (Leung and Chen, 2001). It makes use of the chain-rule property in calculus, together with the transformed space technique, to expand the spatial derivatives.…”

“…The non-uniform discretization is then realized as long as the first factor, ›u/›x, is precisely modeled by the interpolation technique in the second order, as explained in Leung and Chen (2001). For instance, it is first assumed that there exists a set of N points non-uniformly distributed grids over an interval [x min , x max ].…”

Analysis of photonic band‐gap structures in stratified medium

“…As stated in Leung and Chen (2001), it should be noticed that the derivative ›u/›x depends solely on the spatial distribution. Thus, it is necessary to be computed only once, i.e.…”

“…However, the use of FT gives rise to a constraint that the spatial distribution must be kept uniform. Therefore, in order to make the PSTD be flexible as in non-uniform FDTD (Choi and Hoefer, 1986), a nu-PSTD approach is proposed (Leung and Chen, 2001). It makes use of the chain-rule property in calculus, together with the transformed space technique, to expand the spatial derivatives.…”

“…The non-uniform discretization is then realized as long as the first factor, ›u/›x, is precisely modeled by the interpolation technique in the second order, as explained in Leung and Chen (2001). For instance, it is first assumed that there exists a set of N points non-uniformly distributed grids over an interval [x min , x max ].…”

Sub-Nanosecond Electromagnetic-Micromagnetic Dynamic Simulations Using the Finite-Difference Time-Domain Method

“…These field oscillations, even when small, alter the local fields inside the magnetic material and hence yield incorrect magnetisation distributions within magnetic structures. The Gibbs effect may be reduced through the use of non-uniform grids [34] at the expense of increased difficulty in implementation particularly for complex geometries.…”

Pseudospectral time-domain (PSTD) methods for the wave equation: Realising boundary conditions with discrete sine and cosine transforms

“…In acoustics, Kosloff & Kosloff used a cosine basis to solve a one-dimensional wave equation with a Neumann (or sound-hard) boundary condition at each end of the domain [19]. Sine and cosine transforms have also been used in elastodynamics for modelling applied forces that are symmetric or antisymmetric in the plane [20], and in electromagnetics to ensure field components vanish at the boundary, e.g., when modelling wave guides [21,22]. Outside of wave problems, sine and cosine transforms have also been used to impose Dirichlet and Neumann boundary conditions for reaction-diffusion problems [23], to impose slip boundary conditions within fluid simulations [24], to impose symmetric boundary conditions when solving the Elder problem in hydrology [25], to impose Dirichlet boundary conditions when solving the KleinGordonZakharov (KGZ) system [26], and to calculate spectral gradients through symmetric extensions [27].…”

Analysis of photonic band‐gap structures in stratified medium

“…As stated in Leung and Chen (2001), it should be noticed that the derivative ›u/›x depends solely on the spatial distribution. Thus, it is necessary to be computed only once, i.e.…”

“…However, the use of FT gives rise to a constraint that the spatial distribution must be kept uniform. Therefore, in order to make the PSTD be flexible as in non-uniform FDTD (Choi and Hoefer, 1986), a nu-PSTD approach is proposed (Leung and Chen, 2001). It makes use of the chain-rule property in calculus, together with the transformed space technique, to expand the spatial derivatives.…”

“…The non-uniform discretization is then realized as long as the first factor, ›u/›x, is precisely modeled by the interpolation technique in the second order, as explained in Leung and Chen (2001). For instance, it is first assumed that there exists a set of N points non-uniformly distributed grids over an interval [x min , x max ].…”

Sub-Nanosecond Electromagnetic-Micromagnetic Dynamic Simulations Using the Finite-Difference Time-Domain Method

“…These field oscillations, even when small, alter the local fields inside the magnetic material and hence yield incorrect magnetisation distributions within magnetic structures. The Gibbs effect may be reduced through the use of non-uniform grids [34] at the expense of increased difficulty in implementation particularly for complex geometries.…”

Pseudospectral time-domain (PSTD) methods for the wave equation: Realising boundary conditions with discrete sine and cosine transforms

“…In acoustics, Kosloff & Kosloff used a cosine basis to solve a one-dimensional wave equation with a Neumann (or sound-hard) boundary condition at each end of the domain [19]. Sine and cosine transforms have also been used in elastodynamics for modelling applied forces that are symmetric or antisymmetric in the plane [20], and in electromagnetics to ensure field components vanish at the boundary, e.g., when modelling wave guides [21,22]. Outside of wave problems, sine and cosine transforms have also been used to impose Dirichlet and Neumann boundary conditions for reaction-diffusion problems [23], to impose slip boundary conditions within fluid simulations [24], to impose symmetric boundary conditions when solving the Elder problem in hydrology [25], to impose Dirichlet boundary conditions when solving the KleinGordonZakharov (KGZ) system [26], and to calculate spectral gradients through symmetric extensions [27].…”

Analysis of photonic band‐gap structures in stratified medium

“…As stated in Leung and Chen (2001), it should be noticed that the derivative ›u/›x depends solely on the spatial distribution. Thus, it is necessary to be computed only once, i.e.…”

“…However, the use of FT gives rise to a constraint that the spatial distribution must be kept uniform. Therefore, in order to make the PSTD be flexible as in non-uniform FDTD (Choi and Hoefer, 1986), a nu-PSTD approach is proposed (Leung and Chen, 2001). It makes use of the chain-rule property in calculus, together with the transformed space technique, to expand the spatial derivatives.…”

“…The non-uniform discretization is then realized as long as the first factor, ›u/›x, is precisely modeled by the interpolation technique in the second order, as explained in Leung and Chen (2001). For instance, it is first assumed that there exists a set of N points non-uniformly distributed grids over an interval [x min , x max ].…”

Sub-Nanosecond Electromagnetic-Micromagnetic Dynamic Simulations Using the Finite-Difference Time-Domain Method

“…These field oscillations, even when small, alter the local fields inside the magnetic material and hence yield incorrect magnetisation distributions within magnetic structures. The Gibbs effect may be reduced through the use of non-uniform grids [34] at the expense of increased difficulty in implementation particularly for complex geometries.…”

Pseudospectral time-domain (PSTD) methods for the wave equation: Realising boundary conditions with discrete sine and cosine transforms

“…In acoustics, Kosloff & Kosloff used a cosine basis to solve a one-dimensional wave equation with a Neumann (or sound-hard) boundary condition at each end of the domain [19]. Sine and cosine transforms have also been used in elastodynamics for modelling applied forces that are symmetric or antisymmetric in the plane [20], and in electromagnetics to ensure field components vanish at the boundary, e.g., when modelling wave guides [21,22]. Outside of wave problems, sine and cosine transforms have also been used to impose Dirichlet and Neumann boundary conditions for reaction-diffusion problems [23], to impose slip boundary conditions within fluid simulations [24], to impose symmetric boundary conditions when solving the Elder problem in hydrology [25], to impose Dirichlet boundary conditions when solving the KleinGordonZakharov (KGZ) system [26], and to calculate spectral gradients through symmetric extensions [27].…”