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ABSTRACT
The aim of this work is to develop a new scheme for solving the pure advection equation. This scheme formulated within the perturbation finite-difference context not only conserves symplecticity but also preserves the numerical dispersion relation equation. The employed symplectic integrator of second-order accuracy in time enables calculation of a long-time accurate solution in the sense that the Hamiltonian is conserved at all times. The generalized high-order spatially accurate perturbation difference scheme optimizes numerical phase accuracy through the minimization of the difference between the numerical and exact dispersion relation equations. Our proposed new class of phase error reducing perturbation difference schemes can in addition locally capture discontinuities underlying the concept of applying a shope/flux limiter. The high-order spatial accuracy can be recovered in a smooth region. Besides the Fourier analysis of the discretization errors, anisotropy and dispersion analyses are both conducted on the dispersion-relation and symplecticity-preserving pure advection scheme to shed light on the distinguished nature of the proposed scheme. Numerical tests are carried out and the results compare well with the exact solutions, demonstrating the efficiency, accuracy, and the discontinuity-resolving ability using the proposed class of high-resolution perturbation finite-difference schemes.
Nomenclature
| Cr | = | Courant number |
| E | = | error function |
| H | = | Hamiltonian functional |
| k | = | wavenumber |
| p | = | pressure defined in Eq. (25) |
| Re | = | Reynolds number |
| u | = | velocity component in x direction |
| v | = | velocity component in y direction |
| Ve | = | excat phase velocity |
| Vp | = | numerical phase velocity |
| Δt | = | increment time |
| Δx | = | grid spacing |
| θ | = | phase angle |
| λ | = | wavelength |
| φ | = | field variable |
| ψ | = | streamfunction |
| ω | = | angular frequency |
| ωnum | = | numerical angular frequency |
Nomenclature
| Cr | = | Courant number |
| E | = | error function |
| H | = | Hamiltonian functional |
| k | = | wavenumber |
| p | = | pressure defined in Eq. (25) |
| Re | = | Reynolds number |
| u | = | velocity component in x direction |
| v | = | velocity component in y direction |
| Ve | = | excat phase velocity |
| Vp | = | numerical phase velocity |
| Δt | = | increment time |
| Δx | = | grid spacing |
| θ | = | phase angle |
| λ | = | wavelength |
| φ | = | field variable |
| ψ | = | streamfunction |
| ω | = | angular frequency |
| ωnum | = | numerical angular frequency |