Qualitatively accurate spectral schemes for advection and transport
This work addresses the trade-off between conservation properties and convergence speed in numerical methods for transport and continuum equations, offering a solution for applications requiring both accuracy and physical fidelity.
The authors developed a novel spectral discretization technique that conserves key invariants (scalar multiplication and positivity) of transport and continuum equations while maintaining spectral convergence rates, as demonstrated in numerical experiments.
The transport and continuum equations exhibit a number of conservation laws. For example, scalar multiplication is conserved by the transport equation, while positivity of probabilities is conserved by the continuum equation. Certain discretization techniques, such as particle based methods, conserve these properties, but converge slower than spectral discretization methods on smooth data. Standard spectral discretization methods, on the other hand, do not conserve the invariants of the transport equation and the continuum equation. This article constructs a novel spectral discretization technique that conserves these important invariants while simultaneously preserving spectral convergence rates. The performance of this proposed method is illustrated on several numerical experiments.