Parallel computation of rotating flows

Lars Kristian Lundin, Vincent A. Barker, Jens Nørkær Sørensen

    Research output: Contribution to journalJournal articleResearchpeer-review

    Abstract

    This paper deals with the simulation of 3‐D rotating flows based on the velocity‐vorticity formulation of the Navier‐Stokes equations in cylindrical coordinates. The governing equations are discretized by a finite difference method. The solution is advanced to a new time level by a two‐step process. In the first step, the vorticity at the new time level is computed using the velocity at the previous time level. In the second step, the velocity at the new time level is computed using the new vorticity. We discuss here the second part which is by far the most time‐consuming. The numerical problem is that of solving a singular, large, sparse, over‐determined linear system of equations, and the iterative method CGLS is applied for this purpose. We discuss some of the mathematical and numerical aspects of this procedure and report on the performance of our software on a wide range of parallel computers.

    Darbe sprendžiamas trimatis Navje‐Stokso uždavinys, kai lygtys formuluojamos cilindrineje koordinačiu sistemoje, o nežinomaisiais yra greičio komponentes ir sūkurys. Diferencialines lygtys aproksimuojamos baigtiniu skirtumu metodu. Viena algoritmo žingsni sudaro du etapai. Pirmajame etape panaudodami senas greičio komponenčiu reikšmes apskaičiuojame sūkurio reikšme naujuoju laiko momentu. Antrajame etape apskaičiuojamos naujos greičio komponenčiu reikšmes. Straipsnyje didžiausias demesys skiriamas antrajam etapui, kadangi šios algoritmo dalies realizacija reikalauja daugiausia skaičiavimu. Sprendžiama perpildyta tiesiniu lygčiu sistema, kurios matrica yra siguliari, reta ir dideles dimensijos. Naudojamas CGLS iteracinis metodas. Aptariamas lygiagretusis algoritmas ir pateikiami rezultatai skaičiavimo eksperimentu, kurie buvo atlikti su ivairaus tipo lygiagrečiaisiais kompiuteriais.
    Original languageEnglish
    JournalMathematical Modelling and Analysis
    Volume4
    Issue number1
    Pages (from-to)124-134
    ISSN1392-6292
    DOIs
    Publication statusPublished - 1999

    Cite this

    Lundin, Lars Kristian ; Barker, Vincent A. ; Sørensen, Jens Nørkær. / Parallel computation of rotating flows. In: Mathematical Modelling and Analysis. 1999 ; Vol. 4, No. 1. pp. 124-134.
    @article{fc5b52fb259f4292bc27b8421fdda22d,
    title = "Parallel computation of rotating flows",
    abstract = "This paper deals with the simulation of 3‐D rotating flows based on the velocity‐vorticity formulation of the Navier‐Stokes equations in cylindrical coordinates. The governing equations are discretized by a finite difference method. The solution is advanced to a new time level by a two‐step process. In the first step, the vorticity at the new time level is computed using the velocity at the previous time level. In the second step, the velocity at the new time level is computed using the new vorticity. We discuss here the second part which is by far the most time‐consuming. The numerical problem is that of solving a singular, large, sparse, over‐determined linear system of equations, and the iterative method CGLS is applied for this purpose. We discuss some of the mathematical and numerical aspects of this procedure and report on the performance of our software on a wide range of parallel computers.Darbe sprendžiamas trimatis Navje‐Stokso uždavinys, kai lygtys formuluojamos cilindrineje koordinačiu sistemoje, o nežinomaisiais yra greičio komponentes ir sūkurys. Diferencialines lygtys aproksimuojamos baigtiniu skirtumu metodu. Viena algoritmo žingsni sudaro du etapai. Pirmajame etape panaudodami senas greičio komponenčiu reikšmes apskaičiuojame sūkurio reikšme naujuoju laiko momentu. Antrajame etape apskaičiuojamos naujos greičio komponenčiu reikšmes. Straipsnyje didžiausias demesys skiriamas antrajam etapui, kadangi šios algoritmo dalies realizacija reikalauja daugiausia skaičiavimu. Sprendžiama perpildyta tiesiniu lygčiu sistema, kurios matrica yra siguliari, reta ir dideles dimensijos. Naudojamas CGLS iteracinis metodas. Aptariamas lygiagretusis algoritmas ir pateikiami rezultatai skaičiavimo eksperimentu, kurie buvo atlikti su ivairaus tipo lygiagrečiaisiais kompiuteriais.",
    author = "Lundin, {Lars Kristian} and Barker, {Vincent A.} and S{\o}rensen, {Jens N{\o}rk{\ae}r}",
    year = "1999",
    doi = "10.1080/13926292.1999.9637117",
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    Parallel computation of rotating flows. / Lundin, Lars Kristian; Barker, Vincent A.; Sørensen, Jens Nørkær.

    In: Mathematical Modelling and Analysis, Vol. 4, No. 1, 1999, p. 124-134.

    Research output: Contribution to journalJournal articleResearchpeer-review

    TY - JOUR

    T1 - Parallel computation of rotating flows

    AU - Lundin, Lars Kristian

    AU - Barker, Vincent A.

    AU - Sørensen, Jens Nørkær

    PY - 1999

    Y1 - 1999

    N2 - This paper deals with the simulation of 3‐D rotating flows based on the velocity‐vorticity formulation of the Navier‐Stokes equations in cylindrical coordinates. The governing equations are discretized by a finite difference method. The solution is advanced to a new time level by a two‐step process. In the first step, the vorticity at the new time level is computed using the velocity at the previous time level. In the second step, the velocity at the new time level is computed using the new vorticity. We discuss here the second part which is by far the most time‐consuming. The numerical problem is that of solving a singular, large, sparse, over‐determined linear system of equations, and the iterative method CGLS is applied for this purpose. We discuss some of the mathematical and numerical aspects of this procedure and report on the performance of our software on a wide range of parallel computers.Darbe sprendžiamas trimatis Navje‐Stokso uždavinys, kai lygtys formuluojamos cilindrineje koordinačiu sistemoje, o nežinomaisiais yra greičio komponentes ir sūkurys. Diferencialines lygtys aproksimuojamos baigtiniu skirtumu metodu. Viena algoritmo žingsni sudaro du etapai. Pirmajame etape panaudodami senas greičio komponenčiu reikšmes apskaičiuojame sūkurio reikšme naujuoju laiko momentu. Antrajame etape apskaičiuojamos naujos greičio komponenčiu reikšmes. Straipsnyje didžiausias demesys skiriamas antrajam etapui, kadangi šios algoritmo dalies realizacija reikalauja daugiausia skaičiavimu. Sprendžiama perpildyta tiesiniu lygčiu sistema, kurios matrica yra siguliari, reta ir dideles dimensijos. Naudojamas CGLS iteracinis metodas. Aptariamas lygiagretusis algoritmas ir pateikiami rezultatai skaičiavimo eksperimentu, kurie buvo atlikti su ivairaus tipo lygiagrečiaisiais kompiuteriais.

    AB - This paper deals with the simulation of 3‐D rotating flows based on the velocity‐vorticity formulation of the Navier‐Stokes equations in cylindrical coordinates. The governing equations are discretized by a finite difference method. The solution is advanced to a new time level by a two‐step process. In the first step, the vorticity at the new time level is computed using the velocity at the previous time level. In the second step, the velocity at the new time level is computed using the new vorticity. We discuss here the second part which is by far the most time‐consuming. The numerical problem is that of solving a singular, large, sparse, over‐determined linear system of equations, and the iterative method CGLS is applied for this purpose. We discuss some of the mathematical and numerical aspects of this procedure and report on the performance of our software on a wide range of parallel computers.Darbe sprendžiamas trimatis Navje‐Stokso uždavinys, kai lygtys formuluojamos cilindrineje koordinačiu sistemoje, o nežinomaisiais yra greičio komponentes ir sūkurys. Diferencialines lygtys aproksimuojamos baigtiniu skirtumu metodu. Viena algoritmo žingsni sudaro du etapai. Pirmajame etape panaudodami senas greičio komponenčiu reikšmes apskaičiuojame sūkurio reikšme naujuoju laiko momentu. Antrajame etape apskaičiuojamos naujos greičio komponenčiu reikšmes. Straipsnyje didžiausias demesys skiriamas antrajam etapui, kadangi šios algoritmo dalies realizacija reikalauja daugiausia skaičiavimu. Sprendžiama perpildyta tiesiniu lygčiu sistema, kurios matrica yra siguliari, reta ir dideles dimensijos. Naudojamas CGLS iteracinis metodas. Aptariamas lygiagretusis algoritmas ir pateikiami rezultatai skaičiavimo eksperimentu, kurie buvo atlikti su ivairaus tipo lygiagrečiaisiais kompiuteriais.

    U2 - 10.1080/13926292.1999.9637117

    DO - 10.1080/13926292.1999.9637117

    M3 - Journal article

    VL - 4

    SP - 124

    EP - 134

    JO - Mathematical Modelling and Analysis

    JF - Mathematical Modelling and Analysis

    SN - 1392-6292

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