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  • 1. Iaccarino, Gianluca
    et al.
    Pettersson, Per
    Uppsala universitet, Avdelningen för teknisk databehandling.
    Nordström, Jan
    Uppsala universitet, Avdelningen för teknisk databehandling.
    Witteveen, Jeroen
    Numerical methods for uncertainty propagation in high speed flows2010In: Proc. ECCOMAS CFD Conference 2010, Portugal: Tech. Univ. Lisbon , 2010, p. 11-Conference paper (Refereed)
  • 2.
    Pettersson, Per
    Uppsala universitet, Avdelningen för beräkningsvetenskap.
    Uncertainty Quantification and Numerical Methods for Conservation Laws2013Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    Conservation laws with uncertain initial and boundary conditions are approximated using a generalized polynomial chaos expansion approach where the solution is represented as a generalized Fourier series of stochastic basis functions, e.g. orthogonal polynomials or wavelets. The stochastic Galerkin method is used to project the governing partial differential equation onto the stochastic basis functions to obtain an extended deterministic system.

    The stochastic Galerkin and collocation methods are used to solve an advection-diffusion equation with uncertain viscosity. We investigate well-posedness, monotonicity and stability for the stochastic Galerkin system. High-order summation-by-parts operators and weak imposition of boundary conditions are used to prove stability. We investigate the impact of the total spatial operator on the convergence to steady-state. 

    Next we apply the stochastic Galerkin method to Burgers' equation with uncertain boundary conditions. An analysis of the truncated polynomial chaos system presents a qualitative description of the development of the solution over time. An analytical solution is derived and the true polynomial chaos coefficients are shown to be smooth, while the corresponding coefficients of the truncated stochastic Galerkin formulation are shown to be discontinuous. We discuss the problematic implications of the lack of known boundary data and possible ways of imposing stable and accurate boundary conditions.

    We present a new fully intrusive method for the Euler equations subject to uncertainty based on a Roe variable transformation. The Roe formulation saves computational cost compared to the formulation based on expansion of conservative variables. Moreover, it is more robust and can handle cases of supersonic flow, for which the conservative variable formulation fails to produce a bounded solution. A multiwavelet basis that can handle  discontinuities in a robust way is used.

    Finally, we investigate a two-phase flow problem. Based on regularity analysis of the generalized polynomial chaos coefficients, we present a hybrid method where solution regions of varying smoothness are coupled weakly through interfaces. In this way, we couple smooth solutions solved with high-order finite difference methods with non-smooth solutions solved for with shock-capturing methods.

  • 3.
    Pettersson, Per
    et al.
    Department of Mechanical Engineering, Stanford University, USA.
    Abbas, Qaisar
    Uppsala universitet, Avdelningen för teknisk databehandling.
    Iaccarino, Gianluca
    Department of Mechanical Engineering, Stanford University, USA.
    Nordström, Jan
    Uppsala universitet, Avdelningen för teknisk databehandling.
    Efficiency of Shock Capturing Schemes for Burgers' Equation with Boundary Uncertainty2010In: Proc. 7th South African Conference on Computational and Applied Mechanics, South African Association for Theoretical and Applied Mechanics , 2010, p. 22:1-8Conference paper (Other academic)
    Abstract [en]

    Burgers’ equation with uncertain initial and boundary conditions is approximated using a polynomial chaos expansion approach where the solution is represented as a series of stochastic, orthogonal polynomials. Even though the analytical solution is smooth, a number of discontinuities emerge in the truncated system. The solution is highly sensitive to the propagation speed of these discontinuities. High-resolution schemes are needed to accurately capture the behavior of the solution. The emergence of different scales of the chaos modes require dissipation operators to yield accurate solutions. We will compare the results using the MUSCL scheme with previously obtained results using conventional one-sided operators.

  • 4.
    Pettersson, Per
    et al.
    Uppsala universitet, Avdelningen för teknisk databehandling.
    Abbas, Qaisar
    Uppsala universitet, Avdelningen för teknisk databehandling.
    Iaccarino, Gianluca
    Nordström, Jan
    Uppsala universitet, Avdelningen för teknisk databehandling.
    Efficiency of shock capturing schemes for Burgers' equation with boundary uncertainty2010In: Numerical Mathematics and Advanced Applications: 2009, Berlin: Springer-Verlag , 2010, p. 737-745Conference paper (Refereed)
  • 5.
    Pettersson, Per
    et al.
    Uppsala universitet, Avdelningen för teknisk databehandling.
    Iaccarino, Gianluca
    Nordström, Jan
    Uppsala universitet, Avdelningen för teknisk databehandling.
    Boundary procedures for the time-dependent stochastic Burgers' equation2009In: Proc. 19th AIAA CFD Conference, AIAA , 2009Conference paper (Refereed)
  • 6.
    Pettersson, Per
    et al.
    Uppsala universitet, Avdelningen för teknisk databehandling.
    Iaccarino, Gianluca
    Nordström, Jan
    Uppsala universitet, Avdelningen för teknisk databehandling.
    Numerical analysis of Burgers' equation with uncertain boundary conditions using the stochastic Galerkin method2008Report (Other academic)
  • 7.
    Pettersson, Per
    et al.
    Department of Mechanical Engineering, Stanford University, Stanford, USA; Uppsala universitet, Avdelningen för teknisk databehandling.
    Iaccarino, Gianluca
    Department of Mechanical Engineering, Stanford University, Stanford, USA.
    Nordström, Jan
    Uppsala universitet, Avdelningen för teknisk databehandling; Department of Aeronautics and Systems Integration, FOI, Stockholm, Sweden; School of Mechanical, Industrial and Aeronautical Engineering, University of the Witwatersrand, Johannesburg, South Africa.
    Numerical analysis of the Burgers' equation in the presence of uncertainty2009In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 228, p. 8394-8412Article in journal (Refereed)
    Abstract [en]

    The Burgers’ equation with uncertain initial and boundary conditions is investigated usinga polynomial chaos (PC) expansion approach where the solution is represented as a truncatedseries of stochastic, orthogonal polynomials.The analysis of well-posedness for the system resulting after Galerkin projection is presentedand follows the pattern of the corresponding deterministic Burgers equation. Thenumerical discretization is based on spatial derivative operators satisfying the summationby parts property and weak boundary conditions to ensure stability. Similarly to the deterministiccase, the explicit time step for the hyperbolic stochastic problem is proportional tothe inverse of the largest eigenvalue of the system matrix. The time step naturallydecreases compared to the deterministic case since the spectral radius of the continuousproblem grows with the number of polynomial chaos coefficients. An estimate of theeigenvalues is provided.A characteristic analysis of the truncated PC system is presented and gives a qualitativedescription of the development of the system over time for different initial and boundaryconditions. It is shown that a precise statistical characterization of the input uncertainty isrequired and partial information, e.g. the expected values and the variance, are not sufficientto obtain a solution. An analytical solution is derived and the coefficients of the infinitePC expansion are shown to be smooth, while the corresponding coefficients of thetruncated expansion are discontinuous.

  • 8.
    Pettersson, Per
    et al.
    Uppsala universitet, Avdelningen för teknisk databehandling.
    Nordström, Jan
    Uppsala universitet, Avdelningen för teknisk databehandling.
    Iaccarino, Gianluca
    Boundary procedures for the time-dependent Burgers' equation under uncertainty2010In: Acta Mathematica Scientia, ISSN 0252-9602, E-ISSN 1003-3998, Vol. 30, p. 539-550Article in journal (Refereed)
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