20-03-2009/14:00 1 Literature Study Jeroen Wille.

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Presentation transcript:

/14:00 1 Literature Study Jeroen Wille

/14:002 Structure Introduction Scalar advection equation Discontinuous Galerkin Conclusions and further research

/14:003 Introduction Two models Continuous Discontinuous Focus on numerical methods

/14:004 Scalar advection equation 1-dimensional SUPG Perturbation Variable velocity 2-dimensional SUPG Perturbation

/14:005 Scalar advection equation 1D Equation Weak formulation Implicit Euler

/14:006 Scalar advection equation 1D results continuous

/14:007 Scalar advection equation 1D results discontinuous

/14:008 Scalar advection equation 1D Perturbation Perturbation added to equation Neumann boundary condition needed at outflow boundary Extra term to stiffness matrix Extra term to right hand side

/14:009 Scalar advection equation 1D Perturbation results

/14:0010 Scalar advection equation 1D SUPG Extra term p added to test function p is constant per element p is discontinuous over the elements Extra term to mass- and stiffnessmatrix

/14:0011 Scalar advection equation 1D SUPG results

/14:0012 Scalar advection equation 1D variable velocity v no longer fixed, but v = v(x,t) defined by gradient of v equals zero, so v is constant in space Per time step, extra system solved to determine p(1)

/14:0013 Scalar advection equation 1D variable velocity results

/14:0014 Scalar advection equation 2D Equation Weak formulation

/14:0015 Scalar advection equation 2D results continuous

/14:0016 Scalar advection equation 2D results discontinuous

/14:0017 Scalar advection equation 2D Perturbation Perturbation added to equation Neumann boundary condition needed at outflow boundary Extra term to stiffness matrix Extra term to right hand side

/14:0018 Scalar advection equation 2D Perturbation results

/14:0019 Scalar advection equation 2D SUPG Extra term p added to test function p is constant per element p is discontinuous over the elements Extra term to mass- and stiffnessmatrix

/14:0020 Scalar advection equation 2D SUPG results

/14:0021 Discontinuous Galerkin Approximate u per cell by linear combination of polynomials Use legendre-polynomials as basis functions Use upwind flux

/14:0022 Shock detection Detector Near discontinuity, I=O(1) Otherwise I = O(h^(k+2)) So

/14:0023 Limiting Use minmod function Limit in each cell from highest order till zero, or stop when coefficient does not change

/14:0024 Overview

/14:0025 Discontinuous Galerkin results discontinuous

/14:0026 Discontinuous Galerkin results continuous

/14:0027 Conclusions and further research FEM known in 2D, DG not Slope detector sometimes wrongly indicates Look at water flow model Apply numerical methods Obtain realistic data for this model Improve the model