pith. sign in

arxiv: 2605.16684 · v1 · pith:ZM7Y2R4Gnew · submitted 2026-05-15 · 🧮 math.NA · cs.NA

GPU Performance of an Entropy-Stable Discontinuous Galerkin Euler Solver with Non-Conservative Terms

Henry Waterhouse , Maciej Waruszewski , Lucas C. Wilcox , Francis X. Giraldo This is my paper

Pith reviewed 2026-05-19 20:34 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords discontinuous Galerkinentropy stabilityGPU performanceEuler equationsbuoyancynon-conservative termsparallel scaling
0
0 comments X p. Extension
pith:ZM7Y2R4G Add to your LaTeX paper What is a Pith Number? 
\usepackage{pith}
\pithnumber{ZM7Y2R4G}

Prints a linked pith:ZM7Y2R4G badge after your title and writes the identifier into PDF metadata. Compiles on arXiv with no extra files. Learn more

The pith

Entropy-stable discontinuous Galerkin solver for Euler equations with buoyancy reaches nearly 70% of GPU peak performance.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper implements an entropy-stable discontinuous Galerkin method for compressible Euler equations that include buoyancy terms on graphics processing unit hardware. Performance is measured on three-dimensional test problems such as the rising thermal bubble and baroclinic instability in a channel. The GPU code on NVIDIA A100 hardware attains high efficiency through reductions in complex operations and memory traffic, load balancing, and extension of symmetry-based flux savings to the non-symmetric gravity term. These changes yield a factor of ten speedup and more than thirteen times better energy efficiency compared to a highly optimized CPU implementation while preserving entropy stability.

Core claim

By porting the entropy-stable discontinuous Galerkin discretization of the compressible Euler equations with buoyancy to GPU hardware and applying targeted modifications that reduce complex operations, lower memory traffic, improve load balance, and extend symmetry-based flux savings to the non-symmetric gravity term, the solver attains nearly 70% of 64-bit floating-point peak performance for the volume terms kernel on NVIDIA A100 hardware. The resulting implementation delivers a factor of 10 speedup and over 13 times better energy efficiency versus a highly optimized CPU code, achieves the expected doubling of speed when run at 32-bit precision, and shows strong and weak scaling performance

What carries the argument

Symmetry-based flux savings extended to the non-symmetric gravity term within the discontinuous Galerkin volume kernel, which preserves nearly the full factor-of-two speedup for entropy-stable fluxes.

Load-bearing premise

The sequence of GPU-specific modifications preserves both the formal entropy stability and the numerical accuracy of the original discontinuous Galerkin scheme when non-conservative buoyancy terms are present.

What would settle it

Direct side-by-side execution of the GPU and CPU solvers on the rising thermal bubble test case that shows divergence in computed entropy or solution accuracy would disprove preservation of the scheme properties.

Figures

Figures reproduced from arXiv: 2605.16684 by Francis X. Giraldo, Henry Waterhouse, Lucas C. Wilcox, Maciej Waruszewski.

Figure 1
Figure 1. Figure 1: Total volume kernel GPU time per timestep for each cumulative optimization step. Measured on an NVIDIA A100 SXM4 40GB with N = 4, L = 6, 64-bit floating-point precision, using NVIDIA Nsight Compute. 4.2. Single-GPU Our optimized volume kernel is in the compute bound regime for polynomial order N ≥ 3, achieving nearly 70% of the roofline ceiling. We can show this by analyzing a roofline plot of the kernel, … view at source ↗
Figure 2
Figure 2. Figure 2: Roofline plots of the optimized volume kernel on an NVIDIA A100 SXM4 40GB (peak FP64: 9,746 GFLOP/s, peak bandwidth: 1,560 GB/s). Marker shape denotes polynomial order N and color denotes refinement level L (dark to light: coarse to fine). (a) shows every (N, L) configuration; (b) restricts to the most refined mesh available for each polynomial order, isolating the saturated-GPU regime. far [PITH_FULL_IMA… view at source ↗
Figure 3
Figure 3. Figure 3: Percent of total GPU time for the three main kernels at refinement level L = 5. (a): the unoptimized volume kernel dominates for N ≥ 3. (b): after optimization, the surface kernel becomes the dominant cost across the tested polynomial orders [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Single-GPU throughput in thousands of element–time-steps per second as the number of elements grows (K = 1 base mesh refined L times, so element count = 23L). Each curve corresponds to one polynomial order; the flattening of each curve marks the element count at which the GPU is saturated for that order. 9 [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The left panel shows that speedup falls below ideal as GPU count increases due [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Weak-scaling median time per step where the GPU count grows with the refinement level (a single coarse element is subdivided so each GPU holds identical work: L=6 at 8 GPUs, L=7 at 64 GPUs). (a) includes the P=1 baseline; the jump from 1 to 8 GPUs largely reflects the absence of inter-GPU communication in the serial run rather than true weak-scaling loss. (b) baseline shifted to P=8 so only multi-GPU effec… view at source ↗
Figure 7
Figure 7. Figure 7: Slices of simulation results for potential temperature at y = 0 with N = 4, L = 6, and one coarse element in 64-bit (left) and 32-bit (right) precision. This simulation uses the sharp initial perturbation defined in (7). 14 [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Slices of simulation results for potential temperature at y = 0 with N = 4, L = 6, and one coarse element in 64-bit (left) and 32-bit (right) precision. This simulation uses the smooth initial perturbation defined in (8). 15 [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Slice at z = 500 m of pressure (Top) and temperature (Bottom) on day 12 for (a) ESDG and (b) NUMA. Simulation ran with N = 4, L = 3 and Kx = 12, Ky = 2, Kz = 1. As in the rising bubble case, we compare performance in 32-bit and 64-bit precision. We summarize the timing results in [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Slices of simulation results at z = 500 m of pressure at days 0, 6, and 12, in 64-bit (left) and 32-bit (right) precision. Simulation ran with N = 4, L = 3 and Kx = 12, Ky = 2, Kz = 1. 6. Conclusions We presented a GPU implementation of the entropy-stable discontinuous Galerkin method for compressible Euler equations with buoyancy. We demonstrated the performance of the solver on two three-dimensional pro… view at source ↗
Figure 11
Figure 11. Figure 11: Slices of simulation results at z = 500 m of temperature at days 0, 6, and 12 in 64-bit (left) and 32-bit (right) precision. Simulation ran with N = 4, L = 3 and Kx = 12, Ky = 2, Kz = 1 [PITH_FULL_IMAGE:figures/full_fig_p018_11.png] view at source ↗
read the original abstract

The entropy-stable discontinuous Galerkin method for compressible Euler equations with buoyancy is implemented on graphics processing unit (GPU) hardware. We measure the performance of the solver on three-dimensional problems: the rising thermal bubble and the baroclinic instability in a channel. On NVIDIA A100 hardware, the solver achieves nearly 70\% of 64-bit floating-point peak performance for the most computationally expensive kernel (volume terms) and significantly reduces the computational overhead typically incurred by two point entropy-stable fluxes in the volume terms. We also present impressive strong and weak scaling performance of the solver and compare to a highly-optimized central processing unit (CPU) code showing that the GPU kernels are a factor of $10\times$ faster and better than $13\times$ more energy efficient than the CPU code. We also show that the solver achieves the expected $2\times$ speedup when run at 32-bit floating-point peak performance. We discuss the different modifications that we implemented to reach the final form of the GPU implementation and measure the performance gain of each of the implementation strategies ranging from reduction in complex operations and memory traffic as well as load balancing. We also extend symmetry-based flux savings to the non-symmetric gravity term, preserving nearly the full factor-of-two speedup achieved for the symmetric flux.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript presents a GPU implementation of an entropy-stable discontinuous Galerkin solver for the compressible Euler equations with non-conservative buoyancy terms. It evaluates performance on NVIDIA A100 hardware for 3D test cases including the rising thermal bubble and baroclinic instability, reporting nearly 70% of FP64 peak performance for the volume kernel, a 10× speedup and 13× energy efficiency improvement over a CPU reference, nearly 2× speedup at FP32, strong/weak scaling results, and a breakdown of gains from optimizations such as reduced operations, lower memory traffic, load balancing, and extension of symmetry-based flux savings to the non-symmetric gravity term.

Significance. If the GPU kernels preserve the original scheme's entropy stability and accuracy, the work demonstrates that entropy-stable DG methods with non-conservative terms can achieve high fractions of peak GPU performance and substantial speedups over CPU implementations. The per-optimization performance quantification and scaling data would be useful for practitioners developing high-order CFD solvers on accelerators.

major comments (2)
  1. [GPU implementation and flux optimizations] The extension of symmetry-based flux savings to the non-symmetric gravity term (described in the implementation section) is claimed to preserve nearly the full factor-of-two speedup, but the manuscript provides no verification—via discrete entropy inequality checks, comparison of solutions to the CPU reference, or reproduction of the entropy-consistency condition—that the modified two-point volume kernel retains the entropy stability property when buoyancy terms are present. This verification is load-bearing for the claim that the reported performance is achieved by an entropy-stable scheme.
  2. [Performance results and test cases] Performance results (e.g., 70% of peak for the volume kernel, 10× speedup) are presented without error bars, multiple timing runs, or statistical measures of variability. There is also no explicit confirmation that numerical accuracy (error norms or solution fidelity on the test cases) is unchanged after the sequence of GPU modifications including reduced complex operations and load balancing.
minor comments (2)
  1. [Abstract] The abstract states results on 'three-dimensional problems' with two named test cases; clarify whether additional cases were used or if this is a minor wording inconsistency.
  2. [CPU comparison] The CPU baseline is described as 'highly-optimized' but the manuscript does not detail the specific CPU implementation choices, compiler flags, or hardware used for the comparison; this would aid reproducibility of the speedup claims.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed comments on our manuscript. We address each major comment below and agree that the suggested additions will strengthen the paper. Revisions will be incorporated in the next version.

read point-by-point responses

  1. Referee: [GPU implementation and flux optimizations] The extension of symmetry-based flux savings to the non-symmetric gravity term (described in the implementation section) is claimed to preserve nearly the full factor-of-two speedup, but the manuscript provides no verification—via discrete entropy inequality checks, comparison of solutions to the CPU reference, or reproduction of the entropy-consistency condition—that the modified two-point volume kernel retains the entropy stability property when buoyancy terms are present. This verification is load-bearing for the claim that the reported performance is achieved by an entropy-stable scheme.

    Authors: We agree that explicit verification is necessary to support the claim. The extension of the symmetry-based savings to the gravity term is constructed to preserve the two-point entropy-conservative flux properties required by the underlying scheme, but we did not include numerical checks in the submitted manuscript. In the revision we will add discrete entropy inequality verification on the rising thermal bubble and baroclinic instability cases together with direct solution comparisons (including entropy values) against the CPU reference implementation. revision: yes

  2. Referee: [Performance results and test cases] Performance results (e.g., 70% of peak for the volume kernel, 10× speedup) are presented without error bars, multiple timing runs, or statistical measures of variability. There is also no explicit confirmation that numerical accuracy (error norms or solution fidelity on the test cases) is unchanged after the sequence of GPU modifications including reduced complex operations and load balancing.

    Authors: We accept that statistical measures and accuracy confirmation should be provided. The original manuscript reported single-run timings focused on the achieved performance and scaling; we will revise the performance section to include multiple timing runs with error bars or standard deviations. We will also add a direct comparison of numerical accuracy (error norms and solution fidelity) between the CPU reference and the final GPU code after all optimizations to confirm that accuracy is preserved. revision: yes

Circularity Check

0 steps flagged

No circularity: performance results are direct empirical timings against external CPU baseline

full rationale

The paper presents an implementation of a prior entropy-stable DG scheme on GPU hardware, with all central claims consisting of measured wall-clock times, achieved fractions of peak FLOPS, speedup ratios, and energy-efficiency comparisons on A100 hardware versus a CPU reference. These quantities are obtained by direct execution and instrumentation; they do not reduce to any fitted parameter, self-citation loop, or redefinition of the input scheme. The extension of symmetry-based flux savings to the gravity term is described as an implementation choice that preserves the original two-point flux form, but the performance numbers themselves are independent of that choice and are validated by explicit timing breakdowns. No derivation chain collapses by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on standard assumptions of hyperbolic conservation laws and GPU programming models; no new free parameters or invented physical entities are introduced. The only notable domain assumption is that symmetry-based flux reductions remain valid for the non-symmetric gravity term.

axioms (1)
  • domain assumption Symmetry-based flux savings extend to the non-symmetric gravity term while preserving entropy stability and nearly the full factor-of-two reduction in operations.
    Stated in the final sentence of the abstract as an achieved extension.

pith-pipeline@v0.9.0 · 5771 in / 1338 out tokens · 56057 ms · 2026-05-19T20:34:31.502373+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

297 extracted references · 297 canonical work pages · 2 internal anchors

  1. [1]

    , year =

    Betts, John T. , year =. Practical. doi:10.1137/1.9781611976199 , publisher =

    work page doi:10.1137/1.9781611976199

  2. [2]

    SIAM Review , author =

    An. SIAM Review , author =. 2017 , note =. doi:10.1137/16M1062569 , abstract =

    work page doi:10.1137/16m1062569 2017

  3. [3]

    , year =

    Kirk, D.E. , year =. Optimal

    work page

  4. [4]

    Calculus of variations and optimal control theory: a concise introduction , isbn =

    Liberzon, Daniel , year =. Calculus of variations and optimal control theory: a concise introduction , isbn =

    work page

  5. [5]

    , year =

    Hollis, M. , year =. The

    work page

  6. [6]

    Mohamad, A. A. , year =. Lattice. doi:10.1007/978-0-85729-455-5 , language =

    work page doi:10.1007/978-0-85729-455-5

  7. [7]

    , year =

    Pugh, Charles C. , year =. Real. doi:10.1007/978-3-319-17771-7 , language =

    work page doi:10.1007/978-3-319-17771-7

  8. [8]

    , year =

    Giraldo, Francis X. , year =. An. doi:10.1007/978-3-030-55069-1 , language =

    work page doi:10.1007/978-3-030-55069-1

  9. [9]

    and Marsden, J.E

    Chorin, A.J. and Marsden, J.E. , year =. A

    work page

  10. [10]

    and Halperin, Igor and Bilokon, Paul , year =

    Dixon, Matthew F. and Halperin, Igor and Bilokon, Paul , year =. Machine. doi:10.1007/978-3-030-41068-1 , language =

    work page doi:10.1007/978-3-030-41068-1

  11. [11]

    Remarks around 50 lines of

    Alberty, Jochen and Carstensen, Carsten and Funken, Stefan A , file =. Remarks around 50 lines of

    work page

  12. [12]

    Archives of Computational Methods in Engineering , author =

    Eighty. Archives of Computational Methods in Engineering , author =. 2022 , pages =. doi:10.1007/s11831-022-09740-9 , abstract =

    work page doi:10.1007/s11831-022-09740-9 2022

  13. [13]

    Davidson, P. A. , month = dec, year =. Introduction to. doi:10.1017/9781316672853 , language =

    work page doi:10.1017/9781316672853

  14. [14]

    Journal of Computational Physics , author =

    A linearity preserving nodal variation limiting algorithm for continuous. Journal of Computational Physics , author =. 2020 , pages =. doi:10.1016/j.jcp.2020.109390 , abstract =

    work page doi:10.1016/j.jcp.2020.109390 2020

  15. [15]

    Journal of Computational Physics , author =

    Hyperbolic. Journal of Computational Physics , author =. 2002 , pages =. doi:10.1006/jcph.2001.6961 , language =

    work page doi:10.1006/jcph.2001.6961 2002

  16. [16]

    Plasma Astrophysics , author =

    Lecture. Plasma Astrophysics , author =

    work page

  17. [17]

    Toro, E. F. , year =. Riemann solvers and numerical methods for fluid dynamics: a practical introduction , isbn =

    work page

  18. [18]

    Fifty challenging problems in probability with solutions , isbn =

    Mosteller, Frederick , year =. Fifty challenging problems in probability with solutions , isbn =

    work page

  19. [19]

    Nonlinear

    Datseris, George and Parlitz, Ulrich , year =. Nonlinear. doi:10.1007/978-3-030-91032-7 , language =

    work page doi:10.1007/978-3-030-91032-7

  20. [20]

    Chaos and fractals: new frontiers of science , isbn =

    Peitgen, Heinz-Otto and Jürgens, Hartmut and Saupe, Dietmar , year =. Chaos and fractals: new frontiers of science , isbn =

    work page

  21. [21]

    Guide to

    Laaksonen, Antti , year =. Guide to. doi:10.1007/978-3-319-72547-5 , language =

    work page doi:10.1007/978-3-319-72547-5

  22. [22]

    Journal of Scientific Computing , author =

    A. Journal of Scientific Computing , author =. 2022 , pages =. doi:10.1007/s10915-022-01918-4 , abstract =

    work page doi:10.1007/s10915-022-01918-4 2022

  23. [23]

    Journal of Computational Physics , author =

    Ideal. Journal of Computational Physics , author =. 2018 , pages =. doi:10.1016/j.jcp.2018.03.002 , abstract =

    work page doi:10.1016/j.jcp.2018.03.002 2018

  24. [24]

    and Worden, Simon Peter and Hastings, Daniel E

    Ferguson, Dale C. and Worden, Simon Peter and Hastings, Daniel E. , month = sep, year =. The. IEEE Transactions on Plasma Science , publisher =. doi:10.1109/tps.2015.2412775 , abstract =

    work page doi:10.1109/tps.2015.2412775 2015

  25. [25]

    Tóth, Gábor , month = jul, year =. The ∇·. Journal of Computational Physics , publisher =. doi:10.1006/jcph.2000.6519 , language =

    work page doi:10.1006/jcph.2000.6519 2000

  26. [26]

    The Journal of Political Economy , author =

    The. The Journal of Political Economy , author =. 1973 , pages =

    work page 1973

  27. [27]

    Polynomial

    Pettersson, Mass Per and Iaccarino, Gianluca and Nordström, Jan , year =. Polynomial. doi:10.1007/978-3-319-10714-1 , language =

    work page doi:10.1007/978-3-319-10714-1

  28. [28]

    Bulletin of the Polish Academy of Sciences: Technical Sciences , author =

    Numerical solutions of magnetohydrodynamic equations , volume =. Bulletin of the Polish Academy of Sciences: Technical Sciences , author =. 2011 , file =. doi:10.2478/v10175-011-0027-9 , abstract =

    work page doi:10.2478/v10175-011-0027-9 2011

  29. [29]

    Perturbation

    Bush, AlanW , file =. Perturbation

    work page

  30. [30]

    Applied partial differential equations: with

    Haberman, Richard , year =. Applied partial differential equations: with

    work page

  31. [31]

    and Orszag, Steven A

    Bender, Carl M. and Orszag, Steven A. , year =. Advanced. doi:10.1007/978-1-4757-3069-2 , language =

    work page doi:10.1007/978-1-4757-3069-2

  32. [32]

    Nonlinear

    Strogatz, Steven H , month = jan, year =. Nonlinear. doi:10.1201/9780429398490 , language =

    work page doi:10.1201/9780429398490

  33. [33]

    and Cohen, Ira M

    Kundu, Pijush K. and Cohen, Ira M. and Dowling, David R. and Tryggvason, Gretar , year =. Fluid mechanics , isbn =

    work page

  34. [34]

    , year =

    Shreve, Steven E. , year =. Stochastic. doi:10.1007/978-0-387-22527-2 , language =

    work page doi:10.1007/978-0-387-22527-2

  35. [35]

    The SMAI Journal of computational mathematics , author =

    Theoretical and. The SMAI Journal of computational mathematics , author =. 2023 , pages =. doi:10.5802/smai-jcm.95 , abstract =

    work page doi:10.5802/smai-jcm.95 2023

  36. [36]

    Hairer, Ernst and Wanner, Gerhard , year =. Solving. doi:10.1007/978-3-642-05221-7 , language =

    work page doi:10.1007/978-3-642-05221-7

  37. [37]

    Jeevanjee, Nadir , year =. An. doi:10.1007/978-3-319-14794-9 , language =

    work page doi:10.1007/978-3-319-14794-9

  38. [38]

    Principles of mathematical analysis , isbn =

    Rudin, Walter , year =. Principles of mathematical analysis , isbn =

    work page

  39. [39]

    Professor of

    Kunze, Ray , file =. Professor of

    work page

  40. [40]

    Kelley, C. T. , year =. Solving nonlinear equations with iterative methods: solvers and examples in

    work page

  41. [41]

    , year =

    Nahin, Paul J. , year =. Inside. doi:10.1007/978-3-030-43788-6 , language =

    work page doi:10.1007/978-3-030-43788-6

  42. [42]

    Electromagnetism:

    Kamberaj, Hiqmet , year =. Electromagnetism:. doi:10.1007/978-3-030-96780-2 , language =

    work page doi:10.1007/978-3-030-96780-2

  43. [43]

    Journal of Fluid Mechanics , author =

    Compressible magnetoconvection in three dimensions: planforms and nonlinear behaviour , volume =. Journal of Fluid Mechanics , author =. 1995 , pages =. doi:10.1017/S0022112095004630 , abstract =

    work page doi:10.1017/s0022112095004630 1995

  44. [44]

    The Astrophysical Journal , author =

    Flux. The Astrophysical Journal , author =. 1998 , pages =. doi:10.1086/311240 , abstract =

    work page doi:10.1086/311240 1998

  45. [45]

    Schaum's

    Lipschutz, Seymour and Lipson, Marc , year =. Schaum's

    work page

  46. [46]

    Probabilistic

    Krause, Andreas and Hübotter, Jonas , month = feb, year =. Probabilistic. doi:10.48550/arXiv.2502.05244 , abstract =

    work page doi:10.48550/arxiv.2502.05244

  47. [47]

    Mathematical

    Di Pietro, Daniele Antonio and Ern, Alexandre , year =. Mathematical. doi:10.1007/978-3-642-22980-0 , language =

    work page doi:10.1007/978-3-642-22980-0

  48. [48]

    Archives of Computational Methods in Engineering , author =

    A. Archives of Computational Methods in Engineering , author =. 2016 , pages =. doi:10.1007/s11831-015-9152-1 , language =

    work page doi:10.1007/s11831-015-9152-1 2016

  49. [49]

    2011 , file =

    Numerical. 2011 , file =. doi:10.1007/978-3-642-11640-7 , language =

    work page doi:10.1007/978-3-642-11640-7 2011

  50. [50]

    , editor =

    Gander, Martin J. , editor =. 50. Multiple. 2015 , note =. doi:10.1007/978-3-319-23321-5_3 , abstract =

    work page doi:10.1007/978-3-319-23321-5_3 2015

  51. [51]

    Archives of Computational Methods in Engineering , author =

    Current and. Archives of Computational Methods in Engineering , author =. 2019 , pages =. doi:10.1007/s11831-018-9261-8 , abstract =

    work page doi:10.1007/s11831-018-9261-8 2019

  52. [52]

    and Knuth, Donald Ervin and Patashnik, Oren , year =

    Graham, Ronald L. and Knuth, Donald Ervin and Patashnik, Oren , year =. Concrete mathematics: a foundation for computer science , isbn =

    work page

  53. [53]

    SIAM Review , author =

    From. SIAM Review , author =. 2012 , pages =. doi:10.1137/100804036 , abstract =

    work page doi:10.1137/100804036 2012

  54. [54]

    An analysis of the finite element method , language =

    work page

  55. [55]

    Gander, Martin , file =. Time

    work page

  56. [56]

    2000 , file =

    Discontinuous. 2000 , file =. doi:10.1007/978-3-642-59721-3 , language =

    work page doi:10.1007/978-3-642-59721-3 2000

  57. [57]

    and Raviart, P.A

    Lasaint, P. and Raviart, P.A. , year =. On a. Mathematical. doi:10.1016/B978-0-12-208350-1.50008-X , language =

    work page doi:10.1016/b978-0-12-208350-1.50008-x

  58. [58]

    Triangular mesh methods for the neutron transport equation , url =

    Reed, W H and Hill, T R , month = oct, year =. Triangular mesh methods for the neutron transport equation , url =

    work page

  59. [59]

    Journal of Computational Physics , author =

    A finite-element method for the 1-. Journal of Computational Physics , author =. 1982 , pages =. doi:10.1016/0021-9991(82)90107-3 , language =

    work page doi:10.1016/0021-9991(82)90107-3 1982

  60. [60]

    Polynomial approximation of functions and derivatives , language =

    Venturi, Daniele , file =. Polynomial approximation of functions and derivatives , language =

    work page

  61. [61]

    ESAIM: Mathematical Modelling and Numerical Analysis , author =

    The local projection \. ESAIM: Mathematical Modelling and Numerical Analysis , author =. 1989 , pages =. doi:10.1051/m2an/1989230405651 , language =

    work page doi:10.1051/m2an/1989230405651 1989

  62. [62]

    Shu, Bernardo Cockburn Chi-Wang , file =. The

    work page

  63. [63]

    An artificial viscosity approach to high order entropy stable discontinuous

    Chan, Jesse , month = jan, year =. An artificial viscosity approach to high order entropy stable discontinuous. doi:10.48550/arXiv.2501.16529 , abstract =

    work page doi:10.48550/arxiv.2501.16529

  64. [64]

    Applied Numerical Mathematics , author =

    An explicit finite element method for the wave equation , volume =. Applied Numerical Mathematics , author =. 1994 , pages =. doi:10.1016/0168-9274(94)00048-4 , abstract =

    work page doi:10.1016/0168-9274(94)00048-4 1994

  65. [65]

    1989 , pages =

    Journal of Computational Physics , author =. 1989 , pages =. doi:10.1016/0021-9991(89)90183-6 , language =

    work page doi:10.1016/0021-9991(89)90183-6 1989

  66. [66]

    Cockburn, Bernardo and Shu, Chi-Wang , year =

    work page

  67. [67]

    A space-time discontinuous

    Lowrie, R and Roe, N Leer, B, P , month = jun, year =. A space-time discontinuous. 12th. doi:10.2514/6.1995-1658 , language =

    work page doi:10.2514/6.1995-1658 1995

  68. [68]

    Journal of Computational Physics , author =

    Efficient implementation of essentially non-oscillatory shock-capturing schemes , volume =. Journal of Computational Physics , author =. 1988 , pages =. doi:10.1016/0021-9991(88)90177-5 , language =

    work page doi:10.1016/0021-9991(88)90177-5 1988

  69. [69]

    Cockburn, Bernardo and Hou, Suchung and Shu, Chi-Wang , year =. The

    work page

  70. [70]

    Journal of Computational Physics , author =

    On positivity-preserving high order discontinuous. Journal of Computational Physics , author =. 2010 , pages =. doi:10.1016/j.jcp.2010.08.016 , abstract =

    work page doi:10.1016/j.jcp.2010.08.016 2010

  71. [71]

    Journal of Computational Physics , author =

    Entropy stable high order discontinuous. Journal of Computational Physics , author =. 2017 , pages =. doi:10.1016/j.jcp.2017.05.025 , language =

    work page doi:10.1016/j.jcp.2017.05.025 2017

  72. [72]

    Advances in Water Resources , author =

    Positivity-preserving high order well-balanced discontinuous. Advances in Water Resources , author =. 2010 , pages =. doi:10.1016/j.advwatres.2010.08.005 , abstract =

    work page doi:10.1016/j.advwatres.2010.08.005 2010

  73. [73]

    Computer Methods in Applied Mechanics and Engineering , author =

    A space-time finite element method for the wave equation , volume =. Computer Methods in Applied Mechanics and Engineering , author =. 1993 , pages =. doi:10.1016/0045-7825(93)90172-T , language =

    work page doi:10.1016/0045-7825(93)90172-t 1993

  74. [74]

    Journal of Computational Physics , author =

    Space–. Journal of Computational Physics , author =. 2002 , pages =. doi:10.1016/S0021-9991(02)97185-8 , language =

    work page doi:10.1016/s0021-9991(02)97185-8 2002

  75. [75]

    Journal of Computational Physics , author =

    A space–time discontinuous. Journal of Computational Physics , author =. 2013 , pages =. doi:10.1016/j.jcp.2012.08.052 , abstract =

    work page doi:10.1016/j.jcp.2012.08.052 2013

  76. [76]

    and Walters, Gage S

    Frontin, Cory V. and Walters, Gage S. and Witherden, Freddie D. and Lee, Carl W. and Williams, David M. and Darmofal, David L. , month = mar, year =. Foundations of space-time finite element methods: polytopes, interpolation, and integration , shorttitle =. doi:10.48550/arXiv.2012.08701 , abstract =

    work page doi:10.48550/arxiv.2012.08701 2012

  77. [77]

    Computational Mechanics , author =

    Space–time computations in practical engineering applications: a summary of the 25-year history , volume =. Computational Mechanics , author =. 2019 , pages =. doi:10.1007/s00466-018-1620-7 , language =

    work page doi:10.1007/s00466-018-1620-7 2019

  78. [78]

    , year =

    Kopriva, David A. , year =. Implementing. doi:10.1007/978-90-481-2261-5 , language =

    work page doi:10.1007/978-90-481-2261-5

  79. [79]

    Journal of Scientific Computing , author =

    Entropy. Journal of Scientific Computing , author =. 2019 , pages =. doi:10.1007/s10915-019-00933-2 , abstract =

    work page doi:10.1007/s10915-019-00933-2 2019

  80. [80]

    Journal of Computational Physics , author =

    Approximate tensor-product preconditioners for very high order discontinuous. Journal of Computational Physics , author =. 2018 , pages =. doi:10.1016/j.jcp.2017.10.030 , language =

    work page doi:10.1016/j.jcp.2017.10.030 2018

Showing first 80 references.