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A SIMPLE, FAST AND STABILIZED FLOWING FINITE VOLUME METHOD FOR SOLVING GENERAL CURVE EVOLUTION EQUATIONS
, 810
"... Abstract. A new simple Lagrangian method with favorable stability and efficiency properties for computing general plane curve evolutions is presented. The method is based on the flowing finite volume discretization of the intrinsic partial differential equation for updating the position vector of ev ..."
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Abstract. A new simple Lagrangian method with favorable stability and efficiency properties for computing general plane curve evolutions is presented. The method is based on the flowing finite volume discretization of the intrinsic partial differential equation for updating the position vector of evolving family of plane curves. A curve can be evolved in the normal direction by a combination of fourth order terms related to the intrinsic Laplacian of the curvature, second order terms related to the curvature, first order terms related to anisotropy and by a given external velocity field. The evolution is numerically stabilized by an asymptotically uniform tangential redistribution of grid points yielding the first order intrinsic advective terms in the governing system of equations. By using a semiimplicit in time discretization it can be numerically approximated by a solution to linear pentadiagonal systems of equations (in presence of the fourth order terms) or tridiagonal systems (in the case of the second order terms). Various numerical experiments of plane curve evolutions, including, in particular, nonlinear, anisotropic and regularized backward curvature flows, surface diffusion and Willmore flows, are presented and discussed.
3D Embryogenesis Image Segmentation by the Generalized Subjective Surfaces Method using the Finite Volume Technique
"... ABSTRACT. In this paper, an efficient finite volume method for image segmentation is introduced. The method is based on surface evolution governed by a nonlinear PDE, the generalized subjective surfaces equation. Our numerical method is based on semiimplicit time discretization and finite volume sp ..."
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ABSTRACT. In this paper, an efficient finite volume method for image segmentation is introduced. The method is based on surface evolution governed by a nonlinear PDE, the generalized subjective surfaces equation. Our numerical method is based on semiimplicit time discretization and finite volume space approximation. We show examples of image segmentation particularly, we deal with images of early embryogenesis of zebrafish obtained by a confocal microscope. We mention how the segmentation can be useful for analysis of the embryo images and reconstruction of the embryo evolution.
M.: A new level set method for motion in normal direction based on a semiimplicit forwardbackward diffusion approach
 SIAM J. Scientific Computing
, 2010
"... Abstract. We introduce a new level set method for motion in normal direction. It is based on a formulation in the form of a second order forwardbackward diffusion equation. The equation is discretized by the finite volume method. We propose a semiimplicit time discretization taking into account th ..."
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Abstract. We introduce a new level set method for motion in normal direction. It is based on a formulation in the form of a second order forwardbackward diffusion equation. The equation is discretized by the finite volume method. We propose a semiimplicit time discretization taking into account the forward diffusion part of the solution in an implicit way, while the backward diffusion part is treated explicitly. When forward diffusion dominates, a straightforward reconstruction of the solution is used, while larger (smoothing) stencils are used when backward diffusion dominates. The method is precise on coarse grids and is second order accurate for smooth solutions. Numerical experiments show an optimal coupling of time and space steps with τ = h, and no stronger CFL condition is required. Numerical tests with the scheme are discussed on representative examples.
Ohlberger: InflowImplicit/OutflowExplicit Scheme for Solving Advection Equations, in Finite Volumes in Complex Applications VI, Problems & Perspectives, Eds. J.Foˇrt et al
 Proceedings of the Sixth International Conference on Finite Volumes in Complex Applications
"... Abstract We present new method for solving nonstationary advection equations based on the finite volume space discretization and the semiimplicit discretization in time. Its basic idea is that outflow from a cell is treated explicitly while inflow is treated implicitly. Since the matrix of the sys ..."
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Abstract We present new method for solving nonstationary advection equations based on the finite volume space discretization and the semiimplicit discretization in time. Its basic idea is that outflow from a cell is treated explicitly while inflow is treated implicitly. Since the matrix of the system in this new I 2 OE method is determined by the inflow fluxes it is an Mmatrix yielding favourable solvability and stability properties. The method allows large time steps at a fixed spatial grid without losing stability and not deteriorating precision which makes it attractive for practical applications. Our new method is exact for any choice of a discrete time step on uniform rectangular grids in the case of constant velocity transport of quadratic functions in any dimension. We show that it is formally second order accurate in space and time for 1D advection problems with variable velocity and numerical experiments indicates its second order accuracy for smooth solutions in general.
FluxBased Level Set Method for TwoPhase Flow Towards pure finite volume discretization of incompress ible twophase flow using a level set formulation
"... ABSTRACT. We describe a pure finite volume method for the problem of incompressible twophase flow using a level set formulation to track the interface between phases. Similar discrete local balance formulations are used for the approximation of the conservation of all related values — the momentum, ..."
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ABSTRACT. We describe a pure finite volume method for the problem of incompressible twophase flow using a level set formulation to track the interface between phases. Similar discrete local balance formulations are used for the approximation of the conservation of all related values — the momentum, the mass and the level set function. Moreover, a possible jump of the pressure at the interface is modeled directly within the method. As result we found very good conservation properties of the method and negligible parasite currents in examples considered here. The presented calculations are performed without any artificial postprocessing steps often used in the numerical methods for twophase flows based on the level set formulation.
Contents lists available at ScienceDirect Journal of Computational and Applied
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METHODS FOR SOLVING ADVECTION EQUATIONS
"... Abstract. We introduce a new class of methods for solving nonstationary advection equations. The new methods are based on finite volume space discretizations and a semiimplicit discretization in time. Its basic idea is that outflow from a cell is treated explicitly while inflow is treated implicit ..."
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Abstract. We introduce a new class of methods for solving nonstationary advection equations. The new methods are based on finite volume space discretizations and a semiimplicit discretization in time. Its basic idea is that outflow from a cell is treated explicitly while inflow is treated implicitly. This is natural, since we know what is outflowing from a cell at the old time step but we leave the method to resolve a system of equations determined by the inflows to a cell to obtain the solution values at the new time step. The matrix of the system in our inflowimplicit/outflowexplicit (I2OE) method is determined by the inflow fluxes which results in a Mmatrix yielding favourable stability properties for the scheme. Since the explicit (outflow) part is not always dominated by the implicit (inflow) part and thus some oscillations can occur, we build a stabilization based on the upstream weighted averages with coefficients determined by the fluxcorrected transport approach [2, 19] yielding high resolution versions, S1I2OE and S2I2OE, of the basic scheme. We prove that our new method is exact for any choice of a discrete time step on uniform rectangular grids in the case of constant velocity transport of quadratic functions in any dimension. We also show its formal second order accuracy in space and time for 1D advection problems with variable velocity. Although designed for nondivergence free velocity fields, we show that the basic I2OE scheme is locally mass
HIGHRESOLUTION FLUXBASED LEVEL SET METHOD ∗
"... Abstract. A new highresolution fluxbased finite volume method for general advection equations in nondivergent form including a level set equation for moving interfaces is introduced. The method is applicable to the case of nondivergence free velocity and to general unstructured grids in higher dim ..."
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Abstract. A new highresolution fluxbased finite volume method for general advection equations in nondivergent form including a level set equation for moving interfaces is introduced. The method is applicable to the case of nondivergence free velocity and to general unstructured grids in higher dimensions. We show that the method is consistent and that the numerical solution fulfills the discrete minimum/maximum principle. Numerical experiments show its second order accuracy for smooth solutions as well as for solutions with discontinuous derivatives and on general unstructured meshes. Numerical examples for passive transport and shrinking of dynamic interfaces, including examples with topological changes, are presented using locally adapted twodimensional and threedimensional grids.