The sharp CFL condition of the piecewise constant sparse grid discontinuous Galerkin method for high-dimensional transport equations
Abstract
We establish the sharp CFL condition for the piecewise constant sparse grid discontinuous Galerkin (DG) method with forward Euler time stepping, applied to transport equations with constant coefficients on periodic domains in arbitrary dimensions. For the transport velocity c=(c1,…,cd) and a uniform mesh of size h, we prove that the scheme is L2 stable if and only if Δt ≤ h/max1≤ ℓ≤ d|cℓ|, whereas the corresponding full grid upwind scheme is well-known to require Δt≤ h/∑ℓ=1d |cℓ|. The sparse grid discretization therefore enlarges the admissible time step by a factor of (∑ℓ=1d |cℓ|)/(max1≤ ℓ≤ d|cℓ|), which lies between $1$ and d and reaches d for isotropic transport. The proof of sufficiency relies on projection leakage identities for the multilevel Haar decomposition, which allow the mixed directional terms in the energy estimate to be absorbed by the energy discarded by the sparse grid projection. The proof of sharpness follows from alternating modes in one dimension at the finest level. As a by-product, we obtain explicit formulas for the L2 operator norm and the spectral radius of the amplification operator. For spaces over general downward closed index sets, we derive an explicit sufficient CFL condition and a geometric criterion for its sharpness. Numerical experiments in two and four dimensions confirm the theoretical results.
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BibTeX
@misc{huang2026sharp,
title = {{The sharp CFL condition of the piecewise constant sparse grid discontinuous Galerkin method for high-dimensional transport equations}},
author = {Juntao Huang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17312},
eprint = {2609.17312},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17312},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The sharp CFL condition of the piecewise constant sparse grid discontinuous Galerkin method for high-dimensional transport equations
AU - Juntao Huang
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.17312
UR - https://arxiv.org/abs/2609.17312
AB - We establish the sharp CFL condition for the piecewise constant sparse grid discontinuous Galerkin (DG) method with forward Euler time stepping, applied to transport equations with constant coefficients on periodic domains in arbitrary dimensions. For the transport velocity c=(c_1,…,c_d) and a uniform mesh of size h, we prove that the scheme is L^2 stable if and only if Δt ≤ h/max_1≤ ℓ≤ d|c_ℓ|, whereas the corresponding full grid upwind scheme is well-known to require Δt≤ h/∑_ℓ=1^d |c_ℓ|. The sparse grid discretization therefore enlarges the admissible time step by a factor of (∑_ℓ=1^d |c_ℓ|)/(max_1≤ ℓ≤ d|c_ℓ|), which lies between $1$ and d and reaches d for isotropic transport. The proof of sufficiency relies on projection leakage identities for the multilevel Haar decomposition, which allow the mixed directional terms in the energy estimate to be absorbed by the energy discarded by the sparse grid projection. The proof of sharpness follows from alternating modes in one dimension at the finest level. As a by-product, we obtain explicit formulas for the L^2 operator norm and the spectral radius of the amplification operator. For spaces over general downward closed index sets, we derive an explicit sufficient CFL condition and a geometric criterion for its sharpness. Numerical experiments in two and four dimensions confirm the theoretical results.
ER -