{ "id": "ep8ps9jvch", "slug": "shock-capturing-scheme", "title": "Shock-capturing scheme", "updated": "2026-10-21", "topic_path": [ { "id": "physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical" }, { "id": "physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math" }, { "id": "physical.math.analysis", "label": "Analysis and mathematical models", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.analysis" }, { "id": "physical.math.analysis.numerical_analysis_and_computation", "label": "Numerical analysis and computation", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.analysis.numerical_analysis_and_computation" }, { "id": "physical.math.analysis.numerical_analysis_and_computation.discontinuous-galerkin-and-high-order-schemes", "label": "Discontinuous Galerkin and high-order schemes", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.analysis.numerical_analysis_and_computation.discontinuous-galerkin-and-high-order-schemes" } ], "geo": [ { "id": "geo.nongeo.t1946.physical.math.analysis.numerical_analysis_and_computation", "label": "Non-geographic · 1946 to 2000: Numerical analysis and computation", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.analysis.numerical_analysis_and_computation", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1946", "label": "Non-geographic · 1946 to 2000", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946" }, { "id": "geo.nongeo.t1946.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical" }, { "id": "geo.nongeo.t1946.physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math" }, { "id": "geo.nongeo.t1946.physical.math.analysis", "label": "Analysis and mathematical models", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.analysis" }, { "id": "geo.nongeo.t1946.physical.math.analysis.numerical_analysis_and_computation", "label": "Numerical analysis and computation", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.analysis.numerical_analysis_and_computation" } ] } ], "excerpt": "A shock-capturing scheme is a numerical method for hyperbolic conservation laws that resolves shock waves and other discontinuities automatically, smearing them over grid cells instead of tracking the front explicitly.", "snippet": "A shock-capturing scheme is a numerical method for hyperbolic conservation laws that resolves shock waves and other discontinuities automatically, smearing them over grid cells instead of tracking the front explicitly.", "node": "physical.math.analysis.numerical_analysis_and_computation.discontinuous-galerkin-and-high-order-schemes", "markdown": "# Shock-capturing scheme\n\nA shock-capturing scheme is a numerical method for hyperbolic conservation laws that resolves shock waves and other discontinuities automatically, smearing them over one or more grid cells rather than tracking the discontinuity front explicitly.[1](https://doi.org/10.1017/cbo9780511791253) The same discretization is used at every grid point, and regularization is achieved by adding numerical dissipation that prevents or limits Gibbs oscillations near the shock.[2](http://dma.dima.uniroma1.it:8080/users/lsa_gsn/MATERIALE/review.pdf) The motivation is practical: fully resolving a shock in the Navier-Stokes equations would require grids on the scale of a mean free path, about \\( 10^{-7} \\) m for air at standard conditions, so under-resolved capturing is the workable alternative.[3](https://onlinelibrary.wiley.com/doi/10.1155/2012/631276) A successful scheme must implicitly incorporate the correct jump conditions, minimize smearing, and introduce no nonphysical oscillations.[1](https://doi.org/10.1017/cbo9780511791253)\n\n| Key fact | Detail |\n|---|---|\n| What it produces | Discontinuities smeared over one or more grid cells, captured automatically without front tracking[1](https://doi.org/10.1017/cbo9780511791253) |\n| Core mechanism | Conservation form plus artificial dissipation, consistent with the integral form of the equations[4](https://ntrs.nasa.gov/api/citations/19830026406/downloads/19830026406.pdf) |\n| Building block | A Riemann solver, used exactly or approximately at each cell interface in Godunov-type methods[4](https://ntrs.nasa.gov/api/citations/19830026406/downloads/19830026406.pdf) • [25](https://www.uv.es/astrorela/simulacionnumerica/node36.html) |\n| Accuracy barrier | Monotone schemes are at most first-order accurate; TVD schemes restore usable accuracy[2](http://dma.dima.uniroma1.it:8080/users/lsa_gsn/MATERIALE/review.pdf) |\n| Limiter cost | Near a discontinuity the discretization degrades to first-order upwind[5](https://www.kursatkara.com/publication/shock-capturing-schemes/shock-capturing-schemes.pdf) |\n| Known failure | Carbuncle instability, reported in blunt-body computations, and no universally stable flux function[6](https://ar5iv.labs.arxiv.org/html/2308.03428) • [7](https://www.sciencedirect.com/science/article/abs/pii/S0021999120308706) |\n| Main applications | Computational fluid dynamics, computational astronomy and astrophysics, semiconductor device simulation, traffic flow models, and computational biology[8](https://psycnet.apa.org/doi/10.1137/070679065) |\n\n## How it works\n\nThe scheme discretizes the conservation law in integral (conservative) form, because the key to treating shock waves numerically is the combination of conservation and artificial dissipation.[4](https://ntrs.nasa.gov/api/citations/19830026406/downloads/19830026406.pdf) The elementary problem is the [Riemann problem](https://www.edgepedia.org/riemann-problem): the hyperbolic equation with piecewise-constant initial data containing a single jump. Its solution is a [similarity solution](https://www.edgepedia.org/similarity-solution) depending on \\( x/t \\) alone, made of a finite set of waves propagating with constant speeds.[1](https://doi.org/10.1017/cbo9780511791253) Upwind differencing in the style of Godunov's method makes the solution of this shock-tube problem a building block of the difference scheme.[4](https://ntrs.nasa.gov/api/citations/19830026406/downloads/19830026406.pdf)\n\nDissipation can be added explicitly. In modern artificial-viscosity formulations a sensor drives a diffusion term appended to the conservation law,\n\n\\[ \\frac{\\partial u}{\\partial t} + \\nabla \\cdot F(u) = \\nabla \\cdot \\left( \\nu_{\\mathrm{AV}} \\nabla u \\right), \\]\n\nwhere a non-dimensional form of the velocity divergence (dilation) serves as a point-wise sensor triggered around shocks.[9](https://www.mit.edu/~cuongng/project/cfd6/cfd6.pdf) The trade-off is unavoidable: in the original von Neumann-Richtmyer formulation the shock transition overshoots and approaches its final value in an oscillatory fashion, and changing the available parameter diminished the overshoot but never completely eliminated it.[10](https://www.osti.gov/servlets/purl/4244712) The quantitative influence of the added terms can, however, be made as small as desired by choosing a sufficiently fine mesh.[11](https://doi.org/10.1063/1.1699639)\n\n## How it is done\n\nA practitioner works on a finite volume grid. Solving the Riemann problem with left and right states taken from neighboring cell averages yields information used to compute a numerical flux and update the cell averages over a time step.[1](https://doi.org/10.1017/cbo9780511791253) The canonical sequence is:\n\n1. **Reconstruct** left and right interface values from cell averages. A classical route to second-order accuracy uses first-order fluxes together with a conservative reconstruction operator assigning values \\( U_{i+1/2}^{-} \\), \\( U_{i+1/2}^{+} \\).[12](https://www.esaim-proc.org/articles/proc/pdf/2005/02/bouchut.pdf) The MUSCL family of interpolation schemes is the established choice, enforced by a minmod limiter, \\( \\kappa_{1} \\leftarrow \\mathrm{minmod}[\\kappa_{1}, \\kappa_{m}] \\).[13](https://ntrs.nasa.gov/api/citations/20240006106/downloads/Huynh_ICCFD_2024_May_13.pdf)\n2. **Use Riemann solvers at each cell boundary, exactly or approximately**; Godunov's method can be read as a finite volume method doing precisely this, while some other shock-capturing methods, such as artificial-viscosity methods, do not require solving a Riemann problem at each interface.[14](https://www.nas.nasa.gov/assets/nas/pdf/ams/2026/AMS_20260521_Cummings.pdf) • [4](https://ntrs.nasa.gov/api/citations/19830026406/downloads/19830026406.pdf) • [25](https://www.uv.es/astrorela/simulacionnumerica/node36.html)\n3. **Update** the cell averages with the numerical fluxes and advance in time, subject to a stability condition of Courant-Friedrichs-Lewy type; early Los Alamos work tested schemes under a strengthened CFL criterion.[10](https://www.osti.gov/servlets/purl/4244712)\n\n## Origin\n\nAn influential later treatment of finite-volume methods for hyperbolic problems is Randall J. LeVeque's 2002 book *Finite Volume Methods for Hyperbolic Problems*, published by [Cambridge University Press](https://www.edgepedia.org/cambridge-university-press), which presents high-resolution Godunov methods with Riemann solvers and limiters in the CLAWPACK package.[1](https://doi.org/10.1017/cbo9780511791253)\n\nArtificial viscosity modifies the equations of hydrodynamics by adding terms that simplify stepwise solution of shock problems.[15](https://www.osti.gov/biblio/1401360) The concept was classified until 1993, which produced a misattribution of the invention primarily to von Neumann.[15](https://www.osti.gov/biblio/1401360)\n\nGodunov's scheme has a separate lineage. [16](https://ar5iv.labs.arxiv.org/html/0810.0649) A NASA history places the scheme in a PhD dissertation at [Moscow State University](https://www.edgepedia.org/moscow-state-university), published in 1959.[14](https://www.nas.nasa.gov/assets/nas/pdf/ams/2026/AMS_20260521_Cummings.pdf) A two-dimensional extension for nonstationary gas dynamics exists.[17](https://www.mathnet.ru/php/archive.phtml?jrnid=zvmmf&option_lang=eng&paperid=7967&wshow=paper)\n\n## Variants\n\nModern shock-capturing schemes appropriate for computing weak solutions fall into two basic classes, TVD and ENO schemes.[18](https://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1297&context=nasapub) Classical design conditions are monotonicity preservation, total variation diminishment, and monotonicity; because monotone schemes are at most first-order accurate, TVD schemes dominated the 1980s.[2](http://dma.dima.uniroma1.it:8080/users/lsa_gsn/MATERIALE/review.pdf)\n\n**ENO and WENO.** ENO reconstructions use an adaptive stencil that avoids interpolating across discontinuities, guaranteeing \\( \\mathrm{TV}(v^{n+1}) \\le \\mathrm{TV}(v^{n}) + O(h^{r}) \\), though free stencil adaptation caused convergence problems later cured by biasing toward the central stencil.[2](http://dma.dima.uniroma1.it:8080/users/lsa_gsn/MATERIALE/review.pdf) A paper is cited as the starting point of ENO schemes,[19](https://lsec.cc.ac.cn/lcfd/DEWENO/icase-1997-65.pdf) and Harten's 1987 two-dimensional extension appears in the Lecture Notes in [Mathematics](https://www.edgepedia.org/mathematics) record.[20](https://doi.org/10.1007/bfb0078315) WENO schemes build a high-order flux from a convex linear combination of lower-order reconstructions over staggered stencils, with near-zero weights on stencils crossed by a discontinuity; [2](http://dma.dima.uniroma1.it:8080/users/lsa_gsn/MATERIALE/review.pdf) WENO achieves arbitrarily high formal order in smooth regions with stable, nonoscillatory, sharp discontinuity transitions, and requires no user-tuned parameters; it exists in both finite volume and finite difference forms.[8](https://psycnet.apa.org/doi/10.1137/070679065) • [21](https://par.nsf.gov/biblio/10226108-essentially-non-oscillatory-weighted-essentially-non-oscillatory-schemes)\n\n**High-order and limiter-based methods.** Discontinuous Galerkin, spectral, and [flux reconstruction](https://www.edgepedia.org/flux-reconstruction) methods tend to produce oscillations near shocks and are stabilized by limiting procedures such as MUSCL limiting.[13](https://ntrs.nasa.gov/api/citations/20240006106/downloads/Huynh_ICCFD_2024_May_13.pdf) An alternative to TVD's accuracy loss is enforcing monotonicity preservation at the discrete level.[2](http://dma.dima.uniroma1.it:8080/users/lsa_gsn/MATERIALE/review.pdf) On the neural side, the S-PINN framework for weakly hyperbolic systems integrates the Rankine-Hugoniot condition into the network architecture with a mask function to capture discontinuities with high resolution.[22](https://doi.org/10.1088/1402-4896/ae2391)\n\n## Applications\n\nWENO schemes alone are applied in computational fluid dynamics, computational astronomy and astrophysics, semiconductor device simulation, traffic flow models, and computational biology.[8](https://psycnet.apa.org/doi/10.1137/070679065)\n\n## Limitations and alternatives\n\n**Dissipation-oscillation trade-off.** First-order upwind reconstruction preserves monotonicity but is too diffusive.[5](https://www.kursatkara.com/publication/shock-capturing-schemes/shock-capturing-schemes.pdf) Flux limiters resolve this locally: near a discontinuity the discretization becomes first-order upwind, while away from it the limitation is removed and accuracy rises to second or third order for five-point stencil schemes.[5](https://www.kursatkara.com/publication/shock-capturing-schemes/shock-capturing-schemes.pdf) Godunov's first-order method shows no wiggles before or after a steady strong shock but evident smearing.[14](https://www.nas.nasa.gov/assets/nas/pdf/ams/2026/AMS_20260521_Cummings.pdf)\n\n**Failure modes.** The carbuncle phenomenon occurs in supersonic blunt-body computations using the Roe solver, and shock instability affects Godunov-type and kinetic schemes alike.[6](https://ar5iv.labs.arxiv.org/html/2308.03428) Even dissipative solvers suffer shock instabilities when spatial accuracy is raised to fifth order in finite-volume WENO schemes; stable capture of strong shocks requires sufficient dissipation on transverse faces and at least two points within the numerical shock structure perpendicular to the shock.[6](https://ar5iv.labs.arxiv.org/html/2308.03428) Local characteristic decomposition helps mitigate but does not eliminate the instability.[6](https://ar5iv.labs.arxiv.org/html/2308.03428) A study of shock-capturing flux functions for one- and two-dimensional hypersonic shock problems found that none were universally stable across all dimensions and test cases, motivating high-order shock-fitting alternatives.[7](https://www.sciencedirect.com/science/article/abs/pii/S0021999120308706)\n\n**Alternatives.** Shock-fitting treats shocks as genuine discontinuities governed by Rankine-Hugoniot relations and often gives more accurate representations, but is feasible only when the shock topology is extremely simple; shock capturing uses one discretization everywhere.[2](http://dma.dima.uniroma1.it:8080/users/lsa_gsn/MATERIALE/review.pdf) Historically, Lagrangian formulations treated shocks either by shock fitting or by artificial viscosity.[23](https://opensky.ucar.edu/system/files/2024-08/technotes_49.pdf) For small-scale-dependent (nonclassical, undercompressive) shocks, standard capturing must be replaced by front-tracking, finite-difference, or finite-volume schemes that reproduce physically meaningful dissipation mechanisms characterized by a kinetic relation, a family of paths, or an admissible boundary set.[24](https://www.cambridge.org/core/journals/acta-numerica/article/abs/numerical-methods-with-controlled-dissipation-for-smallscale-dependent-shocks/69956E5DD54B36728E92D5E9522B6D70)\n\n## References\n\n1. [Randall J. LeVeque (2002). Finite Volume Methods for Hyperbolic Problems. Cambridge University Press eBooks.](https://doi.org/10.1017/cbo9780511791253)\n2. [Numerical methods for high-speed flows](http://dma.dima.uniroma1.it:8080/users/lsa_gsn/MATERIALE/review.pdf)\n3. [The Role of Mesh Generation, Adaptation, and Refinement on the Computation of Flows Featuring Strong Shocks](https://onlinelibrary.wiley.com/doi/10.1155/2012/631276)\n4. [NASA NTRS document (von Neumann memorial lecture-style review of CFD schemes)](https://ntrs.nasa.gov/api/citations/19830026406/downloads/19830026406.pdf)\n5. [Numerical assessments of high-order accurate shock capturing schemes: Kelvin–Helmholtz type vortical structures in high-resolutions (San & Kara, Computers & Fluids 89, 2014, 254–276)](https://www.kursatkara.com/publication/shock-capturing-schemes/shock-capturing-schemes.pdf)\n6. [Numerical stability analysis of shock-capturing methods for strong shocks II: high-order finite-volume schemes](https://ar5iv.labs.arxiv.org/html/2308.03428)\n7. [A novel stabilization method for high-order shock fitting with finite element methods](https://www.sciencedirect.com/science/article/abs/pii/S0021999120308706)\n8. [High Order Weighted Essentially Nonoscillatory Schemes for Convection Dominated Problems (SIAM Review)](https://psycnet.apa.org/doi/10.1137/070679065)\n9. [Dilation-based shock capturing for high-order methods](https://www.mit.edu/~cuongng/project/cfd6/cfd6.pdf)\n10. [Los Alamos Scientific Laboratory report on difference equations for discontinuous solutions](https://www.osti.gov/servlets/purl/4244712)\n11. [A Method for the Numerical Calculation of Hydrodynamic Shocks (von Neumann & Richtmyer, 1950)](https://doi.org/10.1063/1.1699639)\n12. [An introduction to finite volume methods for hyperbolic conservation laws (Bouchut, ESAIM Proceedings)](https://www.esaim-proc.org/articles/proc/pdf/2005/02/bouchut.pdf)\n13. [Shock Capturing via Limiting for High-Order Methods including Discontinuous Galerkin (Huynh, ICCFD 2024, NASA NTRS)](https://ntrs.nasa.gov/api/citations/20240006106/downloads/Huynh_ICCFD_2024_May_13.pdf)\n14. [NASA AMS briefing: Godunov Implicit Method (Cummings, 2005/2026 presentation)](https://www.nas.nasa.gov/assets/nas/pdf/ams/2026/AMS_20260521_Cummings.pdf)\n15. [Artificial viscosity: back to the basics (OSTI.GOV journal article)](https://www.osti.gov/biblio/1401360)\n16. [Godunov's recollection of the elaboration of his scheme (arXiv 0810.0649)](https://ar5iv.labs.arxiv.org/html/0810.0649)\n17. [S. K. Godunov, A. V. Zabrodin, G. P. Prokopov, 'A computational scheme for two-dimensional nonstationary problems of gas dynamics...' (1961)](https://www.mathnet.ru/php/archive.phtml?jrnid=zvmmf&option_lang=eng&paperid=7967&wshow=paper)\n18. [NASA publication on modern shock-capturing schemes (TVD and ENO classes)](https://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1297&context=nasapub)\n19. [Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws (ICASE Report 97-65, Shu)](https://lsec.cc.ac.cn/lcfd/DEWENO/icase-1997-65.pdf)\n20. [Ami Harten (1987). Preliminary results on the extension of eno schemes to two-dimensional problems. Lecture notes in mathematics.](https://doi.org/10.1007/bfb0078315)\n21. [Essentially non-oscillatory and weighted essentially non-oscillatory schemes (NSF Public Access Repository record)](https://par.nsf.gov/biblio/10226108-essentially-non-oscillatory-weighted-essentially-non-oscillatory-schemes)\n22. [Jain M Francis and colleagues (2025). S-PINN: physics-informed neural networks for solving weakly hyperbolic systems. Physica Scripta.](https://doi.org/10.1088/1402-4896/ae2391)\n23. [NCAR-TN-63-2: A Survey of Difference Methods for Non-Steady Fluid Dynamics](https://opensky.ucar.edu/system/files/2024-08/technotes_49.pdf)\n24. [Numerical methods with controlled dissipation for small-scale dependent shocks (Acta Numerica)](https://www.cambridge.org/core/journals/acta-numerica/article/abs/numerical-methods-with-controlled-dissipation-for-smallscale-dependent-shocks/69956E5DD54B36728E92D5E9522B6D70)\n25. [4.1.3 High-resolution shock-capturing schemes](https://www.uv.es/astrorela/simulacionnumerica/node36.html)\n\n---\n*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Analysis and mathematical models › Numerical analysis and computation › Discontinuous Galerkin and high-order schemes*\n\n*Initially written Sep 29, 2026 · Reviewed: Sep 30, 2026 · Edited: Sep 30, 2026; Oct 21, 2026 · Last review: Sep 30, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [ "https://www.mit.edu/~cuongng/project/cfd6/cfd6.pdf" ], "url": "https://www.edgepedia.org/shock-capturing-scheme", "markdown_url": "https://www.edgepedia.org/shock-capturing-scheme.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. For other uses, organizations over USD 100M in revenue or 100M monthly users license separately.", "spdx": "LicenseRef-Edgepedia-Community-1.0" }, "credit": "\"Shock-capturing scheme\", Edgepedia (EdgeChat), https://www.edgepedia.org/shock-capturing-scheme. Edgepedia Community License 1.0.", "credit_md": "\"[Shock-capturing scheme](https://www.edgepedia.org/shock-capturing-scheme)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/shock-capturing-scheme](https://www.edgepedia.org/shock-capturing-scheme). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).", "credit_html": "\"Shock-capturing scheme\", Edgepedia (EdgeChat), https://www.edgepedia.org/shock-capturing-scheme. Edgepedia Community License 1.0.", "speakable": "A shock-capturing scheme is a numerical method for hyperbolic conservation laws that resolves shock waves and other discontinuities automatically, smearing them over grid cells instead of tracking the front explicitly." }