pith. sign in

arxiv: 2604.22253 · v1 · submitted 2026-04-24 · 🧮 math.NA · cs.NA· math-ph· math.MP

A high order accurate and energy stable continuous Galerkin framework on summation-by-parts form for the incompressible Navier-Stokes equations

Mrityunjoy Mandal , Arnaud G Malan , Prince Nchupang , Jan Nordstr\"om This is my paper

Pith reviewed 2026-05-08 10:45 UTC · model grok-4.3

classification 🧮 math.NA cs.NAmath-phmath.MP
keywords continuous Galerkinsummation-by-partsNavier-Stokes equationsenergy stabilityhigh-order methodsfinite elementsincompressible flowslid-driven cavity
0
0 comments X

The pith

A summation-by-parts continuous Galerkin scheme for the incompressible Navier-Stokes equations achieves high-order accuracy and guaranteed energy stability.

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

This paper develops a continuous Galerkin finite element method for the incompressible Navier-Stokes equations that is formulated using summation-by-parts operators. Boundary conditions are imposed weakly with simultaneous approximation terms, allowing the scheme to handle discontinuous data naturally. The resulting discretization is energy stable by construction, which bounds the numerical solution and prevents blow-up. High-order accuracy up to fourth order is verified through convergence tests, and the method is shown to produce oscillation-free results on lid-driven cavity flows even at points of boundary discontinuity. This combination offers a stable and accurate alternative for simulating incompressible flows without requiring extra stabilization techniques.

Core claim

The central discovery is that by casting the continuous Galerkin discretization of the incompressible Navier-Stokes equations into summation-by-parts form and applying simultaneous approximation term boundary conditions, one obtains a scheme that is provably energy stable while retaining high-order accuracy. Lagrange polynomials of degree up to four are used, and the nonlinear terms are discretized to close the energy estimate. Numerical results confirm fourth-order convergence on manufactured solutions and stable, accurate performance on lid-driven cavity and backward-facing step problems across a range of Reynolds numbers.

What carries the argument

Summation-by-parts operators combined with simultaneous approximation term (SAT) weak boundary conditions within a continuous Galerkin finite element framework, which together ensure the discrete energy estimate holds for the Navier-Stokes system.

If this is right

  • The scheme converges at fourth order for smooth problems as shown by manufactured solution tests.
  • Energy stability is maintained, leading to bounded solutions without artificial dissipation.
  • Discontinuous boundary conditions, such as in lid-driven cavity flow, are handled without oscillations or special treatments.
  • The method applies effectively to both lid-driven cavity and backward-facing step flows at various Reynolds numbers.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The energy stability may allow reliable long-time integrations of incompressible flows.
  • Similar techniques could stabilize high-order discretizations of other nonlinear fluid equations.
  • The approach opens the door to stable high-order simulations in complex geometries with mixed boundary conditions.

Load-bearing premise

The nonlinear convective terms can be discretized using the summation-by-parts operators in a way that allows the energy estimate to close exactly, without needing additional stabilization terms.

What would settle it

Running the fourth-order scheme on the lid-driven cavity problem at high Reynolds number and observing growing kinetic energy or oscillations near the discontinuous corners would falsify the energy stability claim.

Figures

Figures reproduced from arXiv: 2604.22253 by Arnaud G Malan, Jan Nordstr\"om, Mrityunjoy Mandal, Prince Nchupang.

Figure 1
Figure 1. Figure 1: The driven cavity problem geometry (left) and (right) discretization details using 101 × 101 grid points -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 (a) Re = 100 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 (b… view at source ↗
Figure 2
Figure 2. Figure 2: Comparisons of the numerically computed horizontal velocity (𝑢) and vertical velocity (𝑣) profile with the benchmark data (Ghia et al. [23]) along 𝑥 = 0.5 and 𝑦 = 0.5 for different Reynolds numbers Mandal M. et al.: Preprint submitted to Elsevier Page 12 of 19 view at source ↗
Figure 3
Figure 3. Figure 3: Contour plots of speed profiles for different Re Mandal M. et al.: Preprint submitted to Elsevier Page 13 of 19 view at source ↗
Figure 4
Figure 4. Figure 4: Pressure contours at Re = 100 and 10,000 view at source ↗
Figure 5
Figure 5. Figure 5: Schematic representation of backward-facing step geometry with boundary conditions view at source ↗
Figure 6
Figure 6. Figure 6: Contour plots for 𝑢-velocity at Re = 800 view at source ↗
Figure 7
Figure 7. Figure 7: Contour plots for speed at Re = 800 nodal points. We start with zero initial conditions and adopt the same iterative approach outlined previously with a time step size Δ𝑡 = 0.1. To assess the accuracy of our formulation, we compare the numerical results at Re = 800 with the reference results in [22]. Notably, the reference solution was obtained using a much finer mesh with second-order Lagrange polynomials… view at source ↗
Figure 8
Figure 8. Figure 8: Vorticity contour at Re = 800 view at source ↗
Figure 9
Figure 9. Figure 9: Pressure contour at Re = 800 (a) (b) (c) view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of (a) 𝑢-velocity and (b) vorticity profiles at 𝑥 = 7 and 𝑥 = 15 with [22], and (c) 𝑢-velocity variation along the inflow and outflow boundaries compared with the reference data [45] at Re = 800. 7. Summary and Conclusions A high-order CGFEM formulation in the SBP-SAT framework has been developed to solve initial-boundary value problems for the incompressible Navier–Stokes (INS) equations. The … view at source ↗
read the original abstract

This paper presents a high-order accurate Continuous Galerkin Finite Element Method (CGFEM) for solving the initial boundary value problems governed by the Incompressible Navier-Stokes (INS) equations. We discretize the INS equations using the CGFEM approach in Summation-By-Parts (SBP) form. Lagrange polynomials of up to 4th order are employed. The boundary conditions are imposed weakly using the Simultaneous Approximation Term (SAT) technique, which accommodates discontinuous boundary data without special treatment. The resulting SBP-SAT formulation guarantees an energy stable discretization. The efficiency of the proposed framework is demonstrated by solving a series of numerical tests. Initially, the Method of Manufactured Solutions (MMS) is employed to demonstrate 4th order convergence. Subsequently, the 4th order accurate scheme is applied to a classical benchmark problem featuring discontinuous boundary conditions: the lid-driven cavity flow over a wide range of Reynolds numbers. Accurate and oscillation-free solutions are achieved even in the vicinity of the discontinuous top corner boundaries. Lastly, a canonical backward-facing step flow problem is solved, where accuracy and efficiency are demonstrated.

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 / 1 minor

Summary. The manuscript develops a continuous Galerkin finite-element discretization of the incompressible Navier-Stokes equations written in summation-by-parts (SBP) form, employing Lagrange polynomials up to degree 4. Boundary conditions are imposed weakly via the simultaneous approximation term (SAT) technique. The central claim is that the resulting SBP-SAT scheme is energy stable by construction. Fourth-order accuracy is verified on manufactured solutions, after which the method is applied to the lid-driven cavity flow (across a range of Reynolds numbers) and the backward-facing step, where the authors report accurate, oscillation-free solutions even near discontinuous boundary data.

Significance. If the discrete energy estimate is rigorously closed for the nonlinear convective term, the work would supply a high-order, provably stable CG framework for incompressible flows that naturally accommodates discontinuous boundary data without special treatment. The reported fourth-order MMS convergence and practical performance on standard benchmarks would then constitute useful evidence of both accuracy and robustness.

major comments (2)
  1. [Abstract / formulation] Abstract and formulation section: the assertion that the SBP-SAT formulation 'guarantees an energy stable discretization' is load-bearing for the entire contribution, yet no explicit derivation of the discrete energy estimate is supplied. In particular, it is not shown how the chosen continuous Lagrange elements and convective discretization satisfy the SBP inner-product identity (u_h, C_h(u_h, u_h))_h = boundary terms only, nor how the discrete divergence-free condition cancels the pressure contribution. Without these steps the stability guarantee does not follow from the SBP property alone.
  2. [Numerical results] Numerical experiments: while fourth-order convergence is demonstrated via MMS, the manuscript does not report a direct numerical check of the discrete energy balance (e.g., time evolution of the discrete kinetic energy or its dissipation rate) on the lid-driven cavity or backward-facing step. Such a verification would be required to confirm that the nonlinear term closes as claimed under the chosen operators.
minor comments (1)
  1. [Discretization] The description of how the SBP property is realized for standard continuous Lagrange elements (including the specific quadrature or operator construction) should be expanded; standard CG mass and stiffness matrices do not automatically deliver the required summation-by-parts identity for the convective term.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which help clarify the presentation of our energy-stability claims. We address each major point below and will incorporate the suggested clarifications and verifications into the revised manuscript.

read point-by-point responses

  1. Referee: [Abstract / formulation] Abstract and formulation section: the assertion that the SBP-SAT formulation 'guarantees an energy stable discretization' is load-bearing for the entire contribution, yet no explicit derivation of the discrete energy estimate is supplied. In particular, it is not shown how the chosen continuous Lagrange elements and convective discretization satisfy the SBP inner-product identity (u_h, C_h(u_h, u_h))_h = boundary terms only, nor how the discrete divergence-free condition cancels the pressure contribution. Without these steps the stability guarantee does not follow from the SBP property alone.

    Authors: We agree that the current manuscript would benefit from a more self-contained, step-by-step derivation of the discrete energy estimate. In the revised version we will add a dedicated subsection (immediately following the definition of the SBP-SAT operators) that explicitly demonstrates: (i) that the continuous Lagrange basis together with the chosen quadrature yields the required SBP inner-product identity for the convective term, so that (u_h, C_h(u_h,u_h))_h reduces to boundary terms only; and (ii) that the weakly enforced discrete divergence-free condition, combined with the consistent discretization of the pressure gradient, produces exact cancellation of the pressure work term inside the energy balance. This will make the stability argument fully rigorous and independent of external references. revision: yes

  2. Referee: [Numerical results] Numerical experiments: while fourth-order convergence is demonstrated via MMS, the manuscript does not report a direct numerical check of the discrete energy balance (e.g., time evolution of the discrete kinetic energy or its dissipation rate) on the lid-driven cavity or backward-facing step. Such a verification would be required to confirm that the nonlinear term closes as claimed under the chosen operators.

    Authors: We concur that a direct numerical verification of the discrete energy balance on the benchmark problems would strengthen the practical evidence for the theoretical stability result. In the revised manuscript we will include additional figures that plot the time evolution of the discrete kinetic energy (and its dissipation rate) for the lid-driven cavity at Re = 100, 1000 and 5000, as well as for the backward-facing step. These plots will demonstrate that the kinetic energy remains bounded and that the observed dissipation is consistent with the viscous term, thereby confirming that the nonlinear convective contribution closes as predicted by the SBP-SAT analysis. revision: yes

Circularity Check

0 steps flagged

No circularity: energy stability follows from standard SBP-SAT construction without reduction to fitted inputs or self-referential definitions.

full rationale

The paper discretizes INS via CGFEM on SBP form with SAT weak boundary imposition, then states that the resulting formulation guarantees energy stability. This claim rests on the SBP property enabling a discrete energy estimate that mimics the continuous one (including convective term cancellation under appropriate skew or divergence form). The MMS convergence test and lid-driven cavity/backward-facing step benchmarks serve as independent verification rather than tautological confirmation. No equations reduce the stability result to a parameter fit, and any self-citations to prior SBP work are not load-bearing for the central guarantee. The derivation chain remains self-contained against external SBP theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of SBP operators compatible with continuous Galerkin elements and on the closure of the energy estimate for the nonlinear system under weak SAT boundary treatment; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption Summation-by-parts operators exist for the chosen continuous Galerkin basis and element types
    Invoked to obtain the discrete energy estimate from the SBP identity.
  • ad hoc to paper The nonlinear convective term can be discretized so that the energy estimate remains non-positive without additional stabilization
    Required for the stability guarantee to hold for the full incompressible Navier-Stokes system.

pith-pipeline@v0.9.0 · 5518 in / 1330 out tokens · 39104 ms · 2026-05-08T10:45:31.844039+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

71 extracted references · 71 canonical work pages

  1. [1]

    Encapsulatedhighorderdifferenceoperatorsoncurvilinearnon-conforminggrids

    Ålund,O.,Nordström,J.,2019. Encapsulatedhighorderdifferenceoperatorsoncurvilinearnon-conforminggrids. JournalofComputational Physics 385, 209–224

    work page 2019

  2. [2]

    Unified analysis of discontinuous Galerkin methods for elliptic problems

    Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D., 2002. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM journal on numerical analysis 39, 1749–1779

    work page 2002

  3. [3]

    OnahigherorderaccuratefullydiscreteGalerkinapproximationtotheNavier-Stokes equations

    Baker,G.A.,Dougalis,V.A.,Karakashian,O.A.,1982. OnahigherorderaccuratefullydiscreteGalerkinapproximationtotheNavier-Stokes equations. Mathematics of Computation 39, 339–375

    work page 1982

  4. [4]

    Stream function-vorticity driven cavity solution using𝑝finite elements

    Barragy, E., Carey, G., 1997. Stream function-vorticity driven cavity solution using𝑝finite elements. Computers & Fluids 26, 453–468

    work page 1997

  5. [5]

    Experimental and Computational Investigation of Ahmed Body for Ground Vehicle Aerodynamics

    Bayraktar, I., Landman, D., Baysal, O., 2001. Experimental and Computational Investigation of Ahmed Body for Ground Vehicle Aerodynamics. SAE transactions , 321–331

    work page 2001

  6. [6]

    Benchmark spectral results on the lid-driven cavity flow

    Botella, O., Peyret, R., 1998. Benchmark spectral results on the lid-driven cavity flow. Computers & Fluids 27, 421–433

    work page 1998

  7. [7]

    Mixed and Hybrid Finite Element Methods

    Brezzi, F., Fortin, M., 2012. Mixed and Hybrid Finite Element Methods. volume 15. Springer Science & Business Media

    work page 2012

  8. [8]

    Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations

    Brooks, A.N., Hughes, T.J., 1982. Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Computer methods in applied mechanics and engineering 32, 199–259

    work page 1982

  9. [9]

    Isogeometric Analysis: Stable elements for the 2D Stokes equation

    Buffa, A., De Falco, C., Sangalli, G., 2011. Isogeometric Analysis: Stable elements for the 2D Stokes equation. International Journal for Numerical Methods in Fluids 65, 1407–1422

    work page 2011

  10. [10]

    Numerical analysis

    Burden, R.L., Faires, J.D., 2010. Numerical analysis. 9th ed., Brooks/Cole, Cengage Learning

    work page 2010

  11. [11]

    Fictitious domain finite element methods using cut elements: I

    Burman, E., Hansbo, P., 2010. Fictitious domain finite element methods using cut elements: I. A stabilized Lagrange multiplier method. Computer Methods in Applied Mechanics and Engineering 199, 2680–2686

    work page 2010

  12. [12]

    Finite Element technique for optimal pressure recovery from stream function formulation of viscous flows

    Cayco, M., Nicolaides, R., 1986. Finite Element technique for optimal pressure recovery from stream function formulation of viscous flows. Mathematics of computation 46, 371–377

    work page 1986

  13. [13]

    Analysis of nonconforming stream function and pressure Finite Element spaces for the Navier-Stokes equations

    Cayco, M., Nicolaides, R., 1989. Analysis of nonconforming stream function and pressure Finite Element spaces for the Navier-Stokes equations. Computers & Mathematics with Applications 18, 745–760

    work page 1989

  14. [14]

    Journal of Aircraft 56, 344–355

    Changfoot,D.M.,Malan,A.G.,Nordström,J.,2019.HybridComputational-Fluid-DynamicsPlatformtoInvestigateAircraftTrailingVortices. Journal of Aircraft 56, 344–355

    work page 2019

  15. [15]

    A Finite Element formulation for incompressible flow problems using a generalized streamline operator

    Cruchaga, M.A., Oñate, E., 1997. A Finite Element formulation for incompressible flow problems using a generalized streamline operator. Computer Methods in Applied Mechanics and Engineering 143, 49–67

    work page 1997

  16. [16]

    Reviewofsummation-by-parts operatorswithsimultaneousapproximationterms for the numerical solution of partial differential equations

    DelReyFernández, D.C.,Hicken,J.E.,Zingg,D.W., 2014. Reviewofsummation-by-parts operatorswithsimultaneousapproximationterms for the numerical solution of partial differential equations. Computers & Fluids 95, 171–196

    work page 2014

  17. [17]

    Imposing Dirichlet boundary conditions with Nitsche’s method and spline-based finite elements

    Embar, A., Dolbow, J., Harari, I., 2010. Imposing Dirichlet boundary conditions with Nitsche’s method and spline-based finite elements. International journal for numerical methods in engineering 83, 877–898

    work page 2010

  18. [18]

    Errorofthetwo-stepBDFfortheincompressibleNavier-Stokesproblem

    Emmrich,E.,2004. Errorofthetwo-stepBDFfortheincompressibleNavier-Stokesproblem. ESAIM:Modélisationmathématiqueetanalyse numérique 38, 757–764

    work page 2004

  19. [19]

    Isogeometric divergence-conforming B-splines for the steady Navier–Stokes equations

    Evans, J.A., Hughes, T.J., 2013a. Isogeometric divergence-conforming B-splines for the steady Navier–Stokes equations. Mathematical Models and Methods in Applied Sciences 23, 1421–1478

    work page

  20. [20]

    Isogeometric divergence-conforming B-splines for the unsteady Navier–Stokes equations

    Evans, J.A., Hughes, T.J., 2013b. Isogeometric divergence-conforming B-splines for the unsteady Navier–Stokes equations. Journal of Computational Physics 241, 141–167

    work page

  21. [21]

    Numericalcomputationsofviscous,incompressibleflowproblemsusingatwo-levelFiniteElementMethod

    Fairag,F.,2003. Numericalcomputationsofviscous,incompressibleflowproblemsusingatwo-levelFiniteElementMethod. SIAMJournal on Scientific Computing 24, 1919–1929

    work page 2003

  22. [22]

    A test problem for outflow boundary conditions-flow over a backward-facing step

    Gartling, D.K., 1990. A test problem for outflow boundary conditions-flow over a backward-facing step. International Journal for Numerical Methods in Fluids 11, 953–967. 1Any opinion, findings and conclusions or recommendations expressed in this material are those of the author(s) and therefore the NRF do not accept any liability with regard thereto. 2Vie...

    work page 1990

  23. [23]

    High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method

    Ghia, U., Ghia, K.N., Shin, C., 1982. High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method. Journal of computational physics 48, 387–411

    work page 1982

  24. [24]

    Finite Element Approximation of the Navier-Stokes Equations

    Girault, V., Raviart, P.A., 1979. Finite Element Approximation of the Navier-Stokes Equations. volume 749. Springer Berlin

    work page 1979

  25. [25]

    Finite element methods for Navier-Stokes equations: theory and algorithms

    Girault, V., Raviart, P.A., 2012. Finite element methods for Navier-Stokes equations: theory and algorithms. volume 5. Springer Science & Business Media

    work page 2012

  26. [26]

    AfictitiousdomainmethodforDirichletproblemandapplications

    Glowinski,R.,Pan,T.W.,Periaux,J.,1994. AfictitiousdomainmethodforDirichletproblemandapplications. ComputerMethodsinApplied Mechanics and Engineering 111, 283–303

    work page 1994

  27. [27]

    Incompressible Flow and the Finite Element Method, Volume 2: Isothermal Laminar Flow

    Gresho, P., Sani, R., 2000. Incompressible Flow and the Finite Element Method, Volume 2: Isothermal Laminar Flow. Wiley

    work page 2000

  28. [28]

    Some current CFD issues relevant to the incompressible Navier-Stokes equations

    Gresho, P.M., 1991. Some current CFD issues relevant to the incompressible Navier-Stokes equations. Computer Methods in Applied Mechanics and Engineering 87, 201–252

    work page 1991

  29. [29]

    IteratedpenaltymethodsfortheStokesandNavier-Stokesequations

    Gunzburger,M.D.,1989. IteratedpenaltymethodsfortheStokesandNavier-Stokesequations. Finiteelementanalysisinfluids,1040–1045

    work page 1989

  30. [30]

    FiniteElementMethodsforViscousIncompressibleFlows:AGuidetoTheory,Practice,andAlgorithms

    Gunzburger,M.D.,2012. FiniteElementMethodsforViscousIncompressibleFlows:AGuidetoTheory,Practice,andAlgorithms. Elsevier

    work page 2012

  31. [31]

    An unfitted Finite Element Method, based on Nitsche’s method, for elliptic interface problems

    Hansbo, A., Hansbo, P., 2002. An unfitted Finite Element Method, based on Nitsche’s method, for elliptic interface problems. Computer methods in applied mechanics and engineering 191, 5537–5552

    work page 2002

  32. [32]

    Nitsche’s method for coupling non-matching meshes in fluid-structure vibration problems

    Hansbo, P., Hermansson, J., 2003. Nitsche’s method for coupling non-matching meshes in fluid-structure vibration problems. Computational Mechanics 32, 134–139

    work page 2003

  33. [33]

    A Lagrange multiplier method for the finite element solution of elliptic interface problems using non-matching meshes

    Hansbo, P., Lovadina, C., Perugia, I., Sangalli, G., 2005. A Lagrange multiplier method for the finite element solution of elliptic interface problems using non-matching meshes. Numerische Mathematik 100, 91–115

    work page 2005

  34. [34]

    Solving steady-state lid-driven square cavity flows at high Reynolds numbers via a coupled improved element-free Galerkin–reduced integration penalty method

    Hostos, J.C.Á., Bove, J.C.S., Cruchaga, M.A., Fachinotti, V.D., Agelvis, R.A.M., 2021. Solving steady-state lid-driven square cavity flows at high Reynolds numbers via a coupled improved element-free Galerkin–reduced integration penalty method. Computers & Mathematics with Applications 99, 211–228

    work page 2021

  35. [35]

    Finite Element analysis of incompressible viscous flows by the penalty function formulation

    Hughes, T.J., Liu, W.K., Brooks, A., 1979. Finite Element analysis of incompressible viscous flows by the penalty function formulation. Journal of computational physics 30, 1–60

    work page 1979

  36. [36]

    Optimum aerodynamic design using the Navier–Stokes equations

    Jameson, A., Martinelli, L., Pierce, N.A., 1998. Optimum aerodynamic design using the Navier–Stokes equations. Theoretical and computational fluid dynamics 10, 213–237

    work page 1998

  37. [37]

    A Least-Squares Finite Element Method for incompressible Navier-Stokes problems

    Jiang, B.N., 1992. A Least-Squares Finite Element Method for incompressible Navier-Stokes problems. International Journal for Numerical Methods in Fluids 14, 843–859

    work page 1992

  38. [38]

    Initialboundaryvalueproblemsforhyperbolicsystems

    Kreiss,H.O.,1970. Initialboundaryvalueproblemsforhyperbolicsystems. CommunicationsonPureandAppliedMathematics23,277–298

    work page 1970

  39. [39]

    Multiple Hopf bifurcations and flow dynamics inside a 2D singular lid driven cavity

    Lestandi, L., Bhaumik, S., Avatar, G., Azaiez, M., Sengupta, T.K., 2018. Multiple Hopf bifurcations and flow dynamics inside a 2D singular lid driven cavity. Computers & Fluids 166, 86–103

    work page 2018

  40. [40]

    OvernightindustrialLESforexternalaerodynamics

    Löhner,R.,Othmer,C.,Mrosek,M.,Figueroa,A.,Degro,A.,2021. OvernightindustrialLESforexternalaerodynamics. Computers&Fluids 214, 104771

    work page 2021

  41. [41]

    An artificial compressibility CBS method for modelling heat transfer and fluid flow in heterogeneous porous materials

    Malan, A., Lewis, R., 2011. An artificial compressibility CBS method for modelling heat transfer and fluid flow in heterogeneous porous materials. International Journal for Numerical Methods in Engineering 87, 412–423

    work page 2011

  42. [42]

    An improved unsteady, unstructured, artificial compressibility, finite volume scheme for viscous incompressible flows: Part I

    Malan, A., Lewis, R., Nithiarasu, P., 2002a. An improved unsteady, unstructured, artificial compressibility, finite volume scheme for viscous incompressible flows: Part I. Theory and implementation. International Journal for Numerical Methods in Engineering 54, 695–714

    work page

  43. [43]

    An improved unsteady, unstructured, artificial compressibility, finite volume scheme for viscous incompressible flows: Part II

    Malan, A., Lewis, R., Nithiarasu, P., 2002b. An improved unsteady, unstructured, artificial compressibility, finite volume scheme for viscous incompressible flows: Part II. Application. International Journal for Numerical Methods in Engineering 54, 715–729

    work page

  44. [44]

    An SBP-SAT Continuous Galerkin Finite Element Formulation for Smooth and Discontinuous Fields

    Malan, A.G., Nordström, J., 2023. An SBP-SAT Continuous Galerkin Finite Element Formulation for Smooth and Discontinuous Fields. arXiv preprint arXiv:2311.05395

    work page arXiv 2023

  45. [45]

    Weakly viscous two-dimensional incompressible fluid flows using efficient isogeometric finite element method

    Mandal, M., Shaikh, J.H., 2023. Weakly viscous two-dimensional incompressible fluid flows using efficient isogeometric finite element method. Physics of Fluids 35

    work page 2023

  46. [46]

    A novel finite volume discretization method for advection–diffusion systems on stretched meshes

    Merrick, D., Malan, A.G., van Rooyen, J., 2018. A novel finite volume discretization method for advection–diffusion systems on stretched meshes. Journal of Computational Physics 362, 220–242

    work page 2018

  47. [47]

    Blood flow in the cerebral venous system: modeling and simulation

    Miraucourt, O., Salmon, S., Szopos, M., Thiriet, M., 2017. Blood flow in the cerebral venous system: modeling and simulation. Computer methods in biomechanics and biomedical engineering 20, 471–482

    work page 2017

  48. [48]

    Nitsche,J.,1971. Übereinvariationsprinzipzurlösungvondirichlet-problemenbeiverwendungvonteilräumen,diekeinenrandbedingungen unterworfen sind, in: Abhandlungen aus dem mathematischen Seminar der Universität Hamburg, Springer. pp. 9–15

    work page 1971

  49. [49]

    Conservative finite difference formulations, variable coefficients, energy estimates and artificial dissipation

    Nordström, J., 2006. Conservative finite difference formulations, variable coefficients, energy estimates and artificial dissipation. Journal of Scientific Computing 29, 375–404

    work page 2006

  50. [50]

    A roadmap to well posed and stable problems in computational physics

    Nordström, J., 2017. A roadmap to well posed and stable problems in computational physics. Journal of Scientific Computing 71, 365–385

    work page 2017

  51. [51]

    Askew-symmetricenergyandentropystableformulationofthecompressibleEulerequations

    Nordström,J.,2022. Askew-symmetricenergyandentropystableformulationofthecompressibleEulerequations. JournalofComputational Physics 470, 111573

    work page 2022

  52. [52]

    Nonlinear boundary conditions for initial boundary value problems with applications in computational fluid dynamics

    Nordström, J., 2024a. Nonlinear boundary conditions for initial boundary value problems with applications in computational fluid dynamics. Journal of Computational Physics 498, 112685

    work page

  53. [53]

    A skew-symmetric energy stable almost dissipation free formulation of the compressible Navier-Stokes equations

    Nordström, J., 2024b. A skew-symmetric energy stable almost dissipation free formulation of the compressible Navier-Stokes equations. Journal of Computational Physics 512, 113145

    work page

  54. [54]

    Linear and nonlinear boundary conditions: What’s the difference? Journal of Computational Physics , 114649

    Nordström, J., 2025a. Linear and nonlinear boundary conditions: What’s the difference? Journal of Computational Physics , 114649

    work page

  55. [55]

    Open boundary conditions for nonlinear initial boundary value problems

    Nordström, J., 2025b. Open boundary conditions for nonlinear initial boundary value problems. Journal of Computational Physics 530, 113909

    work page

  56. [56]

    Energy stable boundary conditions for the nonlinear incompressible Navier–Stokes equations

    Nordström, J., La Cognata, C., 2019. Energy stable boundary conditions for the nonlinear incompressible Navier–Stokes equations. Mathematics of Computation 88, 665–690. Mandal M. et al.:Preprint submitted to ElsevierPage 18 of 19 Energy stable CGFEM framework on SBP form for INS equations

    work page 2019

  57. [57]

    The spatial operator in the incompressible Navier–Stokes, Oseen and Stokes equations

    Nordström, J., Laurén, F., 2020. The spatial operator in the incompressible Navier–Stokes, Oseen and Stokes equations. Computer Methods in Applied Mechanics and Engineering 363, 112857

    work page 2020

  58. [58]

    An explicit Jacobian for Newton’s method applied to nonlinear initial boundary value problems in summation-by-parts form

    Nordström, J., Laurén, F., Ålund, O., 2024. An explicit Jacobian for Newton’s method applied to nonlinear initial boundary value problems in summation-by-parts form. AIMS mathematics 9, 23291–23312

    work page 2024

  59. [59]

    Amatrix-free,implicit,incompressiblefractional-stepalgorithmforfluid–structureinteractionapplications

    Oxtoby,O.F.,Malan,A.G.,2012. Amatrix-free,implicit,incompressiblefractional-stepalgorithmforfluid–structureinteractionapplications. Journal of Computational Physics 231, 5389–5405

    work page 2012

  60. [60]

    Spectral/ℎ𝑝penalty Least-Squares Finite Element formulation for the steady incompressible Navier–Stokes equations

    Prabhakar, V., Reddy, J., 2006. Spectral/ℎ𝑝penalty Least-Squares Finite Element formulation for the steady incompressible Navier–Stokes equations. Journal of Computational Physics 215, 274–297

    work page 2006

  61. [61]

    Astress-basedLeast-SquaresFinite-ElementmodelforincompressibleNavier–Stokesequations

    Prabhakar,V.,Reddy,J.,2007. Astress-basedLeast-SquaresFinite-ElementmodelforincompressibleNavier–Stokesequations. International journal for numerical methods in fluids 54, 1369–1385

    work page 2007

  62. [62]

    Computational aerodynamics in aircraft design: challenges and opportunities for Euler/Navier-Stokes methods

    Raj, P., Singer, S.W., 1991. Computational aerodynamics in aircraft design: challenges and opportunities for Euler/Navier-Stokes methods. SAE transactions , 2069–2081

    work page 1991

  63. [63]

    Shell finite cell method: a high order fictitious domain approach for thin-walled structures

    Rank, E., Kollmannsberger, S., Sorger, C., Düster, A., 2011. Shell finite cell method: a high order fictitious domain approach for thin-walled structures. Computer Methods in Applied Mechanics and Engineering 200, 3200–3209

    work page 2011

  64. [64]

    Code verification by the Method of Manufactured Solutions

    Roache, P.J., 2002. Code verification by the Method of Manufactured Solutions. J. Fluids Eng. 124, 4–10

    work page 2002

  65. [65]

    Shen,J.,1990.NumericalsimulationoftheregularizeddrivencavityflowsathighReynoldsnumbers.Computermethodsinappliedmechanics and engineering 80, 273–280

    work page 1990

  66. [66]

    The energy method, stability, and nonlinear convection

    Straughan, B., 2013. The energy method, stability, and nonlinear convection. volume 91. Springer Science & Business Media

    work page 2013

  67. [67]

    Grid sensitivity and role of error in computing a lid-driven cavity problem

    Suman, V., Viknesh S, S., Tekriwal, M.K., Bhaumik, S., Sengupta, T.K., 2019. Grid sensitivity and role of error in computing a lid-driven cavity problem. Physical Review E 99, 013305

    work page 2019

  68. [68]

    Review of summation-by-parts schemes for initial–boundary-value problems

    Svärd, M., Nordström, J., 2014. Review of summation-by-parts schemes for initial–boundary-value problems. Journal of Computational Physics 268, 17–38

    work page 2014

  69. [69]

    Computers & fluids 25, 647–665

    Tafti,D.,1996.Comparisonofsomeupwind-biasedhigh-orderformulationswithasecond-ordercentral-differenceschemefortimeintegration of the incompressible Navier-Stokes equations. Computers & fluids 25, 647–665

    work page 1996

  70. [70]

    An efficient second order in time scheme for approximating long time statistical properties of the two dimensional Navier– Stokes equations

    Wang, X., 2012. An efficient second order in time scheme for approximating long time statistical properties of the two dimensional Navier– Stokes equations. Numerische Mathematik 121, 753–779

    work page 2012

  71. [71]

    Hemodynamics of the heart’s left atrium based on a Variational Multiscale-LES numerical method

    Zingaro, A., Menghini, F., Quarteroni, A., et al., 2021. Hemodynamics of the heart’s left atrium based on a Variational Multiscale-LES numerical method. European Journal of Mechanics-B/Fluids 89, 380–400. Mandal M. et al.:Preprint submitted to ElsevierPage 19 of 19

    work page 2021