An official website of the United States government
Abstract Finite volume, weighted essentially non-oscillatory (WENO) schemes require the computation of a smoothness indicator. This can be expensive, especially in multiple space dimensions. We consider the use of the simple smoothness indicator$$\sigma ^{\textrm{S}}= \frac{1}{N_{\textrm{S}}-1}\sum _{j} ({\bar{u}}_{j} - {\bar{u}}_{m})^2$$ , where$$N_{\textrm{S}}$$ is the number of mesh elements in the stencil,$${\bar{u}}_j$$ is the local function average over mesh elementj, and indexmgives the target element. Reconstructions utilizing standard WENO weighting fail with this smoothness indicator. We develop a modification of WENO-Z weighting that gives a reliable and accurate reconstruction of adaptive order, which we denote as SWENOZ-AO. We prove that it attains the order of accuracy of the large stencil polynomial approximation when the solution is smooth, and drops to the order of the small stencil polynomial approximations when there is a jump discontinuity in the solution. Numerical examples in one and two space dimensions on general meshes verify the approximation properties of the reconstruction. They also show it to be about 10 times faster in two space dimensions than reconstructions using the classic smoothness indicator. The new reconstruction is applied to define finite volume schemes to approximate the solution of hyperbolic conservation laws. Numerical tests show results of the same quality as standard WENO schemes using the classic smoothness indicator, but with an overall speedup in the computation time of about 3.5–5 times in 2D tests. Moreover, the computational efficiency (CPU time versus error) is noticeably improved.
more »
« less
This content will become publicly available on April 30, 2026
Spectral Correctness of the Simplicial Discontinuous Galerkin Approximation of the First-Order Form of Maxwell’s Equations with Discontinuous Coefficients
The paper analyzes the discontinuous Galerkin approximation of Maxwell’s equations written in first-order form and with nonhomogeneous magnetic permeability and electric permittivity. Although the Sobolev smoothness index of the solution may be smaller than 1 2 , it is shown that the approximation converges strongly and is therefore spectrally correct. The convergence proof uses the notion of involution and is based on a deflated inf-sup condition and a duality argument. One essential idea is that the smoothness index of the dual solution is always larger than 1 2 irrespective of the regularity of the material properties.
more »
« less
- Award ID(s):
- 2110868
- PAR ID:
- 10640476
- Publisher / Repository:
- SIAM J. NUMER. ANAL., Vol. 63, No. 2, pp. 661–684 Society for Industrial and Applied Mathematics
- Date Published:
- Journal Name:
- SIAM Journal on Numerical Analysis
- Volume:
- 63
- Issue:
- 2
- ISSN:
- 0036-1429
- Page Range / eLocation ID:
- 661 to 684
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
We propose and analyze a two-scale finite element method for the Isaacs equation. The fine scale is given by the mesh size h whereas the coarse scale ε is dictated by an integro-differential approximation of the partial differential equation. We show that the method satisfies the discrete maximum principle provided that the mesh is weakly acute. This, in conjunction with weak operator consistency of the finite element method, allows us to establish convergence of the numerical solution to the viscosity solution as ε , h → 0, and ε ≳ ( h |log h |) 1/2 . In addition, using a discrete Alexandrov Bakelman Pucci estimate we deduce rates of convergence, under suitable smoothness assumptions on the exact solution.more » « less
-
null (Ed.)In this paper, we study superconvergence properties of the ultraweak-local discontinuous Galerkin (UWLDG) method in Tao et al. [To appear in Math. Comput. DOI: https://doi.org/10.1090/mcom/3562 (2020).] for an one-dimensional linear fourth-order equation. With special initial discretizations, we prove the numerical solution of the semi-discrete UWLDG scheme superconverges to a special projection of the exact solution. The order of this superconvergence is proved to be k + min(3, k ) when piecewise ℙ k polynomials with k ≥ 2 are used. We also prove a 2 k -th order superconvergence rate for the cell averages and for the function values and derivatives of the UWLDG approximation at cell boundaries. Moreover, we prove superconvergence of ( k + 2)-th and ( k + 1)-th order of the function values and the first order derivatives of the UWLDG solution at a class of special quadrature points, respectively. Our proof is valid for arbitrary non-uniform regular meshes and for arbitrary k ≥ 2. Numerical experiments verify that all theoretical findings are sharp.more » « less
-
We consider the problem of determining the manifold $$n$$-widths of Sobolev and Besov spaces with error measured in the $$L_p$$-norm. The manifold widths control how efficiently these spaces can be approximated by general non-linear parametric methods with the restriction that the parameter selection and parameterization maps must be continuous. Existing upper and lower bounds only match when the Sobolev or Besov smoothness index $$q$$ satisfies $$q\leq p$$ or $$1 \leq p \leq 2$$. We close this gap and obtain sharp lower bounds for all $$1 \leq p,q \leq \infty$$ for which a compact embedding holds. A key part of our analysis is to determine the exact value of the manifold widths of finite dimensional $$\ell^M_q$$-balls in the $$\ell_p$$-norm when $$p\leq q$$. Although this result is not new, we provide a new proof and apply it to lower bounding the manifold widths of Sobolev and Besov spaces. Our results show that the Bernstein widths, which are typically used to lower bound the manifold widths, decay asymptotically faster than the manifold widths in many cases.more » « less
-
The transport of energetic particles in response to solar wind turbulence is important for space weather. To understand charged particle transport, it is usually assumed that the phase of the turbulence is randomly distributed (the random phase approximation) in quasi-linear theory and simulations. In this paper, we calculate the coherence index, C ϕ , of solar wind turbulence observed by the Helios 2 and Parker Solar Probe spacecraft using the surrogate data technique to check if the assumption is valid. Here, values of C ϕ = 0 and 1 indicate that the phase coherence is random and correlated, respectively. We estimate that the coherence index at the resonant scale of energetic ions (10 MeV protons) is 0.1 at 0.87 and 0.65 au, 0.18 at 0.29 au, and 0.3 (0.35) at 0.09 au for super (sub)-Alfvénic intervals, respectively. Since the random phase approximation corresponds to C ϕ = 0, this may indicate that the random phase approximation is not valid for the transport of energetic particles in the inner heliosphere, especially very close to the Sun ( ∼ 0.09 au).more » « less
