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  • 1.
    La Cognata, Cristina
    Linköpings universitet, Matematiska institutionen, Beräkningsmatematik. Linköpings universitet, Tekniska fakulteten.
    High order summation-by-parts based approximations for discontinuous and nonlinear problems2017Doktoravhandling, med artikler (Annet vitenskapelig)
    Abstract [en]

    Numerical approximations using high order finite differences on summation-byparts (SBP) form are investigated for discontinuous and fully nonlinear systems of partial differential equations. Stability and conservation properties of the approximations are obtained through a weak imposition of interface and boundary conditions with the simultaneous-approximation-term (SAT) technique. The SBP-SAT approximations replicate the continuous integration by parts rule. From this property, well-posedness and integral properties of the continuous problem are mimicked, and energy estimates leading to stability are obtained.

    The first part of the thesis focuses on the simulations of discontinuous linear advection problems. An artificial interface is introduced, separating parts of the spatial domain characterized by different wave speeds. A set of flexible stability conditions at the interface are derived, which can be adapted to yield conservative or non-conservative approximations. This model can be interpreted as a simplified version of nonlinear problems involving jumps at shocks, or as a prototypical of wave propagation through different materials.

    In the second part of the thesis, the vorticity/stream function formulation of the nonlinear momentum equation for an incompressible inviscid fluid is considered. SBP operators are used to derive a new Arakawa-like Jacobian with mimetic properties by combining different consistent approximations of the convection terms. Energy and enstrophy conservation is obtained for periodic problems using schemes with arbitrarily high order of accuracy. These properties are crucial for long-term numerical calculations in climate and weather forecasts or ocean circulation predictions.

    The third and final contribution of the thesis is dedicated to the incompressible Navier-Stokes problem. First, different completely general formulations of energy bounding boundary conditions are derived for the nonlinear equations. The boundary conditions can be used at both far field and solid wall boundaries. The discretisation in time and space with weakly imposed initial and boundary conditions using the SBP-SAT framework is proved to be stable and the divergence free condition is approximated with the design order of the scheme. Next, the same formulations are considered in a linearised setting, whereupon the spectra associated with the initial boundary value problem and its SBP-SAT discretisation are derived using the Laplace-Fourier technique. The influence of different boundary conditions on the spectrum and in particular the convergence to steady state is studied.

  • 2.
    La Cognata, Cristina
    et al.
    Linköpings universitet, Matematiska institutionen, Beräkningsmatematik. Linköpings universitet, Tekniska högskolan.
    Nordström, Jan
    Linköpings universitet, Matematiska institutionen, Beräkningsmatematik. Linköpings universitet, Tekniska högskolan.
    Well-posedness, Stability and Conservation for a Discontinuous Interface Problem2015Rapport (Annet vitenskapelig)
    Abstract [en]

    The advection equation is studied in a completely general two domain setting with different wave-speeds and a time-independent jump-condition at the interface separating the domains. Well-posedness and conservation criteria are derived for the initial-boundary-value problem. The equations are semidiscretized using afinite dfference method on summation-by-parts (SBP) form. The stability and conservation properties of the approximation are studied when the boundary and interface conditions are weakly imposed by the simultaneous approximation term (SAT) procedure. Numerical simulations corroborate the theoretical finndings.

  • 3.
    La Cognata, Cristina
    et al.
    Linköpings universitet, Matematiska institutionen, Beräkningsmatematik. Linköpings universitet, Tekniska fakulteten.
    Nordström, Jan
    Linköpings universitet, Matematiska institutionen, Beräkningsmatematik. Linköpings universitet, Tekniska fakulteten.
    Well-posedness, stability and conservation for a discontinuous interface problem2016Inngår i: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 56, nr 2, s. 681-704Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    The advection equation is studied in a completely general two domain setting with different wave-speeds and a time-independent jump-condition at the interface separating the domains. Well-posedness and conservation criteria are derived for the initial-boundary-value problem. The equations are semi-discretized using a finite difference method on Summation-By-Part (SBP) form. The relation between the stability and conservation properties of the approximation are studied when the boundary and interface conditions are weakly imposed by the Simultaneous-Approximation-Term (SAT) procedure. Numerical simulations corroborate the theoretical findings.

  • 4.
    La Cognata, Cristina
    et al.
    Linköpings universitet, Matematiska institutionen, Beräkningsmatematik. Linköpings universitet, Tekniska fakulteten.
    Nordström, Jan
    Linköpings universitet, Matematiska institutionen, Beräkningsmatematik. Linköpings universitet, Tekniska fakulteten.
    Well-Posedness, Stability and Conservation for a Discontinuous Interface Problem: An Initial Investigation2015Inngår i: Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2014, Springer, 2015, s. 147-155Kapittel i bok, del av antologi (Fagfellevurdert)
    Abstract [en]

    A robust interface treatment for the discontinuous coefficient advection equation satisfying time-independent jump conditions is presented. The aim of the investigation is to show how the different concepts like well-posedness, conservation and stability are related. The equations are discretized using high order finite difference methods on Summation By Parts (SBP) form. The interface conditions are weakly imposed using the Simultaneous Approximation Term (SAT) procedure. Spectral analysis and numerical simulations corroborate the theoretical findings.

  • 5.
    Nordström, Jan
    et al.
    Linköpings universitet, Matematiska institutionen, Beräkningsmatematik. Linköpings universitet, Tekniska fakulteten.
    La Cognata, Cristina
    Linköpings universitet, Matematiska institutionen, Beräkningsmatematik. Linköpings universitet, Tekniska fakulteten.
    Energy Stable Boundary Conditions for the Nonlinear Incompressible Navier-Stokes Equations2017Rapport (Annet vitenskapelig)
    Abstract [en]

    The nonlinear incompressible Navier-Stokes equations with boundary conditions at far fields and solid walls is considered. Two different formulations of boundary conditions are derived using the energy method. Both formulations are implemented in both strong and weak form and lead to an estimate of the velocity field. Equipped with energy bounding boundary conditions, the problem is approximated by using difference operators on summation-by-parts form and weak boundary and initial conditions. By mimicking the continuous analysis, the resulting semi-discrete as well as fully discrete scheme are shown to be provably stable, divergence free and high-order accurate.

  • 6.
    Nordström, Jan
    et al.
    Linköpings universitet, Matematiska institutionen, Beräkningsmatematik. Linköpings universitet, Tekniska fakulteten.
    La Cognata, Cristina
    Linköpings universitet, Matematiska institutionen, Beräkningsmatematik. Linköpings universitet, Tekniska fakulteten.
    Energy Stable Boundary Conditions for the Nonlinear Incompressible Navier-Stokes Equations2019Inngår i: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 88, nr 316, s. 665-690Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    The nonlinear incompressible Navier-Stokes equations with different types of boundary conditions at far fields and solid walls is considered. Two different formulations of boundary conditions are derived using the energy method. Both formulations are implemented in both strong and weak form and lead to an estimate of the velocity field.

    Equipped with energy bounding boundary conditions, the problem is approximated by using discrete derivative operators on summation-by-parts form and weak boundary and initial conditions. By mimicking the continuous analysis, the resulting semi-discrete as well as fully discrete scheme are shown to be provably stable, divergence free, and high-order accurate.

  • 7.
    Sorgentone, Chiara
    et al.
    Department of Mathematics, Royal Institute of Technology (KTH), SE-100 44 Stockholm, Sweden.
    La Cognata, Cristina
    Linköpings universitet, Matematiska institutionen, Beräkningsmatematik. Linköpings universitet, Tekniska högskolan.
    Nordström, Jan
    Linköpings universitet, Matematiska institutionen, Beräkningsmatematik. Linköpings universitet, Tekniska högskolan.
    A New High Order Energy and Enstrophy Conserving Arakawa-like Jacobian Differential Operator2015Rapport (Annet vitenskapelig)
    Abstract [en]

    A new high order Arakawa-like method for the incompressible vorticity equation in two-dimensions has been developed. Mimetic properties such as skewsymmetry, energy and enstrophy conservations for the semi-discretization are proved for periodic problems using arbitrary high order summation-by-partsoperators. Numerical simulations corroborate the theoreticalfindings.

  • 8.
    Sorgentone, Chiara
    et al.
    Department of Mathematics, Royal Institute of Technology (KTH), SE-100 44 Stockholm, Sweden.
    La Cognata, Cristina
    Linköpings universitet, Matematiska institutionen, Beräkningsmatematik. Linköpings universitet, Tekniska fakulteten.
    Nordström, Jan
    Linköpings universitet, Matematiska institutionen, Beräkningsmatematik. Linköpings universitet, Tekniska fakulteten.
    A new high order energy and enstrophy conserving Arakawa-like Jacobian differential operator2015Inngår i: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 301, s. 167-177Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    A new high order Arakawa-like method for the incompressible vorticity equation in two-dimensions has been developed. Mimetic properties such as skew-symmetry, energy and enstrophy conservations for the semi-discretization are proved for periodic problems using arbitrary high order summation-by-parts operators. Numerical simulations corroborate the theoretical findings.

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