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  • 1. Order onlineBuy this publication >>
    Nikkar, Samira
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Stable High Order Finite Difference Methods for Wave Propagation and Flow Problems on Deforming Domains2016Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    We construct stable, accurate and efficient numerical schemes for wave propagation and flow problems posed on spatial geometries that are moving, deforming, erroneously described or non-simply connected. The schemes are on Summation-by-Parts (SBP) form, combined with the Simultaneous Approximation Term (SAT) technique for imposing initial and boundary conditions. The main analytical tool is the energy method, by which well-posedness, stability and conservation are investigated. To handle the deforming domains, time-dependent coordinate transformations are used to map the problem to fixed geometries.

    The discretization is performed in such a way that the Numerical Geometric Conservation Law (NGCL) is satisfied. Additionally, even though the schemes are constructed on fixed domains, time-dependent penalty formulations are necessary, due to the originally moving boundaries. We show how to satisfy the NGCL and present an automatic formulation for the penalty operators, such that the correct number of boundary conditions are imposed, when and where required.

    For problems posed on erroneously described geometries, we investigate how the accuracy of the solution is affected. It is shown that the inaccurate geometry descriptions may lead to wrong wave speeds, a misplacement of the boundary condition, the wrong boundary operator or a mismatch of data. Next, the SBP-SAT technique is extended to time-dependent coupling procedures for deforming interfaces in hyperbolic problems. We prove conservation and stability and show how to formulate the penalty operators such that the coupling procedure is automatically adjusted to the variations of the interface location while the NGCL is preserved.

    Moreover, dual consistent SBP-SAT schemes for the linearized incompressible Navier-Stokes equations posed on deforming domains are investigated. To simplify the derivations of the dual problem and incorporate the motions of the boundaries, the second order formulation is reduced to first order and the problem is transformed to a fixed domain. We prove energy stability and dual consistency. It is shown that the solution as well as the divergence of the solution converge with the design order of accuracy, and that functionals of the solution are superconverging.

    Finally, initial boundary value problems posed on non-simply connected spatial domains are investigated. The new formulation increases the accuracy of the scheme by minimizing the use of multi-block couplings. In order to show stability, the spectrum of the semi-discrete SBP-SAT formulation is studied. We show that the eigenvalues have the correct sign, which implies stability, in combination with the SBP-SAT technique in time.

    List of papers
    1. Fully discrete energy stable high order finite difference methods for hyperbolic problems in deforming domains
    Open this publication in new window or tab >>Fully discrete energy stable high order finite difference methods for hyperbolic problems in deforming domains
    2015 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 291, p. 82-98Article in journal (Refereed) Published
    Abstract [en]

    A time-dependent coordinate transformation of a constant coefficient hyperbolic system of equations which results in a variable coefficient system of equations is considered. By applying the energy method, well-posed boundary conditions for the continuous problem are derived. Summation-by-Parts (SBP) operators for the space and time discretization, together with a weak imposition of boundary and initial conditions using Simultaneously Approximation Terms (SATs) lead to a provable fully-discrete energy-stable conservative finite difference scheme. We show how to construct a time-dependent SAT formulation that automatically imposes boundary conditions, when and where they are required. We also prove that a uniform flow field is preserved, i.e. the Numerical Geometric Conservation Law (NGCL) holds automatically by using SBP-SAT in time and space. The developed technique is illustrated by considering an application using the linearized Euler equations: the sound generated by moving boundaries. Numerical calculations corroborate the stability and accuracy of the new fully discrete approximations.

    Place, publisher, year, edition, pages
    Elsevier, 2015
    Keywords
    Deforming domain; Initial boundary value problems; High order accuracy; Well-posed boundary conditions; Summation-by-parts operators; Stability; Convergence; Conservation; Numerical geometric conservation law; Euler equation; Sound propagation
    National Category
    Mathematics
    Identifiers
    urn:nbn:se:liu:diva-117360 (URN)10.1016/j.jcp.2015.02.027 (DOI)000352230500006 ()
    Available from: 2015-04-24 Created: 2015-04-24 Last updated: 2017-12-04
    2. Hyperbolic systems of equations posed on erroneous curved domains
    Open this publication in new window or tab >>Hyperbolic systems of equations posed on erroneous curved domains
    2016 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 308, p. 438-442Article in journal (Refereed) Published
    Abstract [en]

    The effect of an inaccurate geometry description on the solution accuracy of a hyperbolic problem is discussed. The inaccurate geometry can for example come from an imperfect CAD system, a faulty mesh generator, bad measurements or simply a misconception.

    We show that inaccurate geometry descriptions might lead to the wrong wave speeds, a misplacement of the boundary conditions, to the wrong boundary operator and a mismatch of boundary data.

    The errors caused by an inaccurate geometry description may affect the solution more than the accuracy of the specific discretization techniques used. In extreme cases, the order of accuracy goes to zero. Numerical experiments corroborate the theoretical results.

    Place, publisher, year, edition, pages
    Elsevier, 2016
    Keywords
    Hyperbolic systems; Erroneous curved domains; Inaccurate data; Convergence rate
    National Category
    Mathematics
    Identifiers
    urn:nbn:se:liu:diva-123918 (URN)10.1016/j.jcp.2015.12.048 (DOI)000369086700021 ()
    Available from: 2016-01-13 Created: 2016-01-13 Last updated: 2017-11-30Bibliographically approved
    3. A fully discrete, stable and conservative summation-by-parts formulation for deforming interfaces
    Open this publication in new window or tab >>A fully discrete, stable and conservative summation-by-parts formulation for deforming interfaces
    2016 (English)Report (Other academic)
    Abstract [en]

    We introduce an interface/coupling procedure for hyperbolic problems posedon time-dependent curved multi-domains. First, we transform the problem from Cartesian to boundary-conforming curvilinear coordinates and apply the energy method to derive well-posed and conservative interface conditions.

    Next, we discretize the problem in space and time by employing finite difference operators that satisfy a summation-by-parts rule. The interface condition is imposed weakly using a penalty formulation. We show how to formulate the penalty operators such that the coupling procedure is automatically adjusted to the movements and deformations of the interface, while both stability and conservation conditions are respected.

    The developed techniques are illustrated by performing numerical experiments on the linearized Euler equations and the Maxwell equations. The results corroborate the stability and accuracy of the fully discrete approximations.

    Place, publisher, year, edition, pages
    Linköping: Linköping University Electronic Press, 2016. p. 38
    Series
    LiTH-MAT-R, ISSN 0348-2960 ; 2016:9
    Keywords
    Finite difference, High order accuracy, Deforming domains, Time-dependent interface, Well-posedness, Conservation, Summation-by-parts, Stability, Hyperbolic problems
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:liu:diva-130583 (URN)LiTH-MAT-R--2016/09--SE (ISRN)
    Available from: 2016-08-17 Created: 2016-08-17 Last updated: 2017-04-27Bibliographically approved
    4. A dual consistent summation-by-parts formulation for the linearized incompressible Navier-Stokes equations posed on deforming domains
    Open this publication in new window or tab >>A dual consistent summation-by-parts formulation for the linearized incompressible Navier-Stokes equations posed on deforming domains
    2016 (English)Report (Other academic)
    Abstract [en]

    In this article, well-posedness and dual consistency of the linearized incompressible Navier-Stokes equations posed on time-dependent spatial domains are studied. To simplify the derivation of the dual problem, the second order formulation is transformed to rst order form. Boundary conditions that simultaneously lead to well-posedness of the primal and dual problems are derived.

    We construct fully discrete nite di erence schemes on summation-byparts form, in combination with the simultaneous approximation technique. We prove energy stability and discrete dual consistency. Moreover, we show how to construct the penalty operators such that the scheme automatically adjusts to the variations of the spatial domain, and as a result, stability and discrete dual consistency follow simultaneously.

    The method is illustrated by considering a deforming time-dependent spatial domain in two dimensions. The numerical calculations are performed using high order operators in space and time. The results corroborate the stability of the scheme and the accuracy of the solution. We also show that linear functionals are superconverging. Additionally, we investigate the convergence of non-linear functionals and the divergence of the solution.

    Place, publisher, year, edition, pages
    Linköping: Linköping University Electronic Press, 2016. p. 26
    Series
    LiTH-MAT-R, ISSN 0348-2960 ; 2016:10
    Keywords
    Incompressible Navier-Stokes equations, Deforming domain, Stability, Dual consistency, High order accuracy, Superconvergence
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:liu:diva-130584 (URN)LiTH-MAT-R--2016/10--SE (ISRN)
    Available from: 2016-08-17 Created: 2016-08-17 Last updated: 2016-09-19Bibliographically approved
    5. Summation-by-parts operators for non-simply connected domains
    Open this publication in new window or tab >>Summation-by-parts operators for non-simply connected domains
    2016 (English)Report (Other academic)
    Abstract [en]

    We construct fully discrete stable and accurate numerical schemes for solving partial differential equations posed on non-simply connected spatial domains. The schemes are constructed using summation-by-parts operators in combination with a weak imposition of initial and boundary conditions using the simultaneous approximation term technique.

    In the theoretical part, we consider the two dimensional constant coefficient advection equation posed on a rectangular spatial domain with a hole. We construct the new scheme and study well-posedness and stability. Once the theoretical development is done, the technique is extended to more complex non-simply connected geometries.

    Numerical experiments corroborate the theoretical results and show the applicability of the new approach and its advantages over the standard multi-block technique. Finally, an application using the linearized Euler equations for sound propagation is presented.

    Place, publisher, year, edition, pages
    Linköping: Linköping University Electronic Press, 2016. p. 32
    Series
    LiTH-MAT-R, ISSN 0348-2960 ; 2016:11
    Keywords
    Initial boundary value problems, Stability, Well-posedness, Boundary conditions, Non-simply connected domains, Complex geometries
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:liu:diva-130585 (URN)LiTH-MAT-R--2016/11--SE (ISRN)
    Available from: 2016-08-17 Created: 2016-08-17 Last updated: 2016-08-31Bibliographically approved
    Download full text (pdf)
    Stable High Order Finite Difference Methods forWave Propagation and Flow Problems on Deforming Domains
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  • 2.
    Nikkar, Samira
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    A dual consistent summation-by-parts formulation for the linearized incompressible Navier-Stokes equations posed on deforming domains2016Report (Other academic)
    Abstract [en]

    In this article, well-posedness and dual consistency of the linearized incompressible Navier-Stokes equations posed on time-dependent spatial domains are studied. To simplify the derivation of the dual problem, the second order formulation is transformed to rst order form. Boundary conditions that simultaneously lead to well-posedness of the primal and dual problems are derived.

    We construct fully discrete nite di erence schemes on summation-byparts form, in combination with the simultaneous approximation technique. We prove energy stability and discrete dual consistency. Moreover, we show how to construct the penalty operators such that the scheme automatically adjusts to the variations of the spatial domain, and as a result, stability and discrete dual consistency follow simultaneously.

    The method is illustrated by considering a deforming time-dependent spatial domain in two dimensions. The numerical calculations are performed using high order operators in space and time. The results corroborate the stability of the scheme and the accuracy of the solution. We also show that linear functionals are superconverging. Additionally, we investigate the convergence of non-linear functionals and the divergence of the solution.

    Download full text (pdf)
    Dual consistent summation-by-parts formulation for the linearized incompressible Navier-Stokes equations posed on deforming domains
  • 3.
    Nikkar, Samira
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    A dual consistent summation-by-parts formulation for the linearized incompressible Navier-Stokes equations posed on deforming domains2019In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 376, p. 26p. 322-338Article in journal (Refereed)
    Abstract [en]

    In this article, well-posedness and dual consistency of the linearized constant coefficient incompressible Navier–Stokes equations posed on time-dependent spatial domains are studied. To simplify the derivation of the dual problem and improve the accuracy of gradients, the second order formulation is transformed to first order form. Boundary conditions that simultaneously lead to boundedness of the primal and dual problems are derived.Fully discrete finite difference schemes on summation-by-parts form, in combination with the simultaneous approximation technique, are constructed. We prove energy stability and discrete dual consistency and show how to construct the penalty operators such that the scheme automatically adjusts to the variations of the spatial domain. As a result of the aforementioned formulations, stability and discrete dual consistency follow simultaneously.The method is illustrated by considering a deforming time-dependent spatial domain in two dimensions. The numerical calculations are performed using high order operators in space and time. The results corroborate the stability of the scheme and the accuracy of the solution. We also show that linear functionals are superconverging. Additionally, we investigate the convergence of non-linear functionals and the divergence of the solution.

    Download full text (pdf)
    fulltext
  • 4.
    Nikkar, Samira
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    A fully discrete, stable and conservative summation-by-parts formulation for deforming interfaces2016Report (Other academic)
    Abstract [en]

    We introduce an interface/coupling procedure for hyperbolic problems posedon time-dependent curved multi-domains. First, we transform the problem from Cartesian to boundary-conforming curvilinear coordinates and apply the energy method to derive well-posed and conservative interface conditions.

    Next, we discretize the problem in space and time by employing finite difference operators that satisfy a summation-by-parts rule. The interface condition is imposed weakly using a penalty formulation. We show how to formulate the penalty operators such that the coupling procedure is automatically adjusted to the movements and deformations of the interface, while both stability and conservation conditions are respected.

    The developed techniques are illustrated by performing numerical experiments on the linearized Euler equations and the Maxwell equations. The results corroborate the stability and accuracy of the fully discrete approximations.

    Download full text (pdf)
    Version 1. Fully discrete, stable and conservative summation-by-parts formulation for deforming interfaces
  • 5.
    Nikkar, Samira
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
    Energy Stable High Order Finite Difference Methods for Hyperbolic Equations in Moving Coordinate Systems2013In: AIAA Aerospace Sciences - Fluid Sciences Event, 2013, p. 1-14Conference paper (Other academic)
    Abstract [en]

    A time-dependent coordinate transformation of a constant coeffcient hyperbolic equation which results in a variable coeffcient problem is considered. By using the energy method, we derive well-posed boundary conditions for the continuous problem. It is shown that the number of boundary conditions depend on the coordinate transformation. By using Summation-by-Parts (SBP) operators for the space discretization and weak boundary conditions, an energy stable finite dieffrence scheme is obtained. We also show how to construct a time-dependent penalty formulation that automatically imposes the right number of boundary conditions. Numerical calculations corroborate the stability and accuracy of the approximations.

    Download full text (pdf)
    fulltext
  • 6.
    Nikkar, Samira
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
    Fully Discrete Energy Stable High Order Finite Difference Methods for Hyperbolic Problems in Deforming Domains2014Report (Other academic)
    Abstract [en]

    A time-dependent coordinate transformation of a constant coefficient hyperbolic system of equations which results in a variable coecient system of equations is considered. By applying the energy method, well-posed boundary conditions for the continuous problem are derived. Summation-by-Parts (SBP) operators for the space and time discretization, together with a weak imposition of boundary and initial conditions using Simultaneously Approximation Terms (SATs) lead to a provable fully-discrete energy-stable finite difference scheme.   We show how to construct a time-dependent SAT formulation that automatically imposes boundary conditions, when and where they are required. We also prove that a uniform flow field is preserved, i.e. the numerical Geometric Conservation Law holds automatically by using SBP-SAT in time. The developed technique is illustrated by considering an application using the linearized Euler equations: the sound generated by moving boundaries. Numerical calculations corroborate the stability and accuracy of the new fully discrete approximations.

    Download full text (pdf)
    Fully Discrete Energy Stable High Order Finite Difference Methods for Hyperbolic Problems in Deforming Domains
  • 7.
    Nikkar, Samira
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
    Fully discrete energy stable high order finite difference methods for hyperbolic problems in deforming domains2015In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 291, p. 82-98Article in journal (Refereed)
    Abstract [en]

    A time-dependent coordinate transformation of a constant coefficient hyperbolic system of equations which results in a variable coefficient system of equations is considered. By applying the energy method, well-posed boundary conditions for the continuous problem are derived. Summation-by-Parts (SBP) operators for the space and time discretization, together with a weak imposition of boundary and initial conditions using Simultaneously Approximation Terms (SATs) lead to a provable fully-discrete energy-stable conservative finite difference scheme. We show how to construct a time-dependent SAT formulation that automatically imposes boundary conditions, when and where they are required. We also prove that a uniform flow field is preserved, i.e. the Numerical Geometric Conservation Law (NGCL) holds automatically by using SBP-SAT in time and space. The developed technique is illustrated by considering an application using the linearized Euler equations: the sound generated by moving boundaries. Numerical calculations corroborate the stability and accuracy of the new fully discrete approximations.

    Download full text (pdf)
    fulltext
  • 8.
    Nikkar, Samira
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Fully Discrete Energy Stable High Order Finite Difference Methods for Hyperbolic Problems in Deforming Domains: An initial investigation2015In: Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2014 / [ed] Mejdi Azaïez, Henda El Fekih, Jan S. Hesthaven, Springer, 2015, p. 385-395Chapter in book (Refereed)
    Abstract [en]

    A time-dependent coordinate transformation of a constant coefficient hyperbolic system of equations is considered. We use the energy method to derive well-posed boundary conditions for the continuous problem. Summation-by-Parts (SBP) operators together with a weak imposition of the boundary and initial conditions using Simultaneously Approximation Terms (SATs) guarantee energy-stability of the fully discrete scheme. We construct a time-dependent SAT formulation that automatically imposes the boundary conditions, and show that the numerical Geometric Conservation Law (GCL) holds. Numerical calculations corroborate the stability and accuracy of the approximations. As an application we study the sound propagation in a deforming domain using the linearized Euler equations.

    Download full text (pdf)
    Fully Discrete Energy Stable High Order Finite Difference Methods for Hyperbolic Problems in Deforming Domains
  • 9.
    Nikkar, Samira
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Summation-by-parts operators for non-simply connected domains2016Report (Other academic)
    Abstract [en]

    We construct fully discrete stable and accurate numerical schemes for solving partial differential equations posed on non-simply connected spatial domains. The schemes are constructed using summation-by-parts operators in combination with a weak imposition of initial and boundary conditions using the simultaneous approximation term technique.

    In the theoretical part, we consider the two dimensional constant coefficient advection equation posed on a rectangular spatial domain with a hole. We construct the new scheme and study well-posedness and stability. Once the theoretical development is done, the technique is extended to more complex non-simply connected geometries.

    Numerical experiments corroborate the theoretical results and show the applicability of the new approach and its advantages over the standard multi-block technique. Finally, an application using the linearized Euler equations for sound propagation is presented.

    Download full text (pdf)
    fulltext
  • 10.
    Nikkar, Samira
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Summation-by-Parts Operators for Non-Simply Connected Domains2018In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 40, no 3, p. 1250-1273Article in journal (Refereed)
    Abstract [en]

    We construct fully discrete stable and accurate numerical schemes for solving partial differential equations posed on non-simply connected spatial domains. The schemes are constructed using summation-by-parts operators in combination with a weak imposition of initial and boundary conditions using the simultaneous approximation term technique. In the theoretical part, we consider the two-dimensional constant coefficient advection equation posed on a rectangular spatial domain with a hole. We construct the new scheme and study well-posedness and stability. Once the theoretical development is done, the technique is extended to more complex non-simply connected geometries. Numerical experiments corroborate the theoretical results and show the applicability of the new approach and its advantages over the standard multiblock technique. Finally, an application using the linearized Euler equations for sound propagation is presented.

    Download full text (pdf)
    fulltext
  • 11.
    Nordström, Jan
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Nikkar, Samira
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Hyperbolic systems of equations posed on erroneous curved domains2016In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 308, p. 438-442Article in journal (Refereed)
    Abstract [en]

    The effect of an inaccurate geometry description on the solution accuracy of a hyperbolic problem is discussed. The inaccurate geometry can for example come from an imperfect CAD system, a faulty mesh generator, bad measurements or simply a misconception.

    We show that inaccurate geometry descriptions might lead to the wrong wave speeds, a misplacement of the boundary conditions, to the wrong boundary operator and a mismatch of boundary data.

    The errors caused by an inaccurate geometry description may affect the solution more than the accuracy of the specific discretization techniques used. In extreme cases, the order of accuracy goes to zero. Numerical experiments corroborate the theoretical results.

    Download full text (pdf)
    Fulltext
  • 12.
    Nordström, Jan
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
    Wahlsten, Markus
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
    Nikkar, Samira
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
    Boundary Conditions for Hyperbolic Systems of Equations on Curved Domains2014Report (Other academic)
    Abstract [en]

    Our focus in this paper is on the fundamental system of partial differential equation with boundary conditions (the continuous problem) that all types of numerical methods must respect. First, a constant coefficient hyperbolic system of equations which turns into a variable coefficient system of equations by transforming to a non-cartesian domain is considered. We discuss possible formulations of time-dependent boundary conditions leading to well-posed or strongly well-posed problems. Next, we re-use the previous theoretical derivations for the problem with boundary conditions applied at the wrong position and/or with an incorrect normal (a typical result with a less than perfect mesh generator). Possible error sources are discussed and a crude error estimate is derived.

    Download full text (pdf)
    Boundary Conditions for Hyperbolic Systems of Equations on Curved Domains
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