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[00782] Recent Advances on Mimetic Difference Methods

  • Session Date & Time : 1C (Aug.21, 13:20-15:00)
  • Type : Proposal of Minisymposium
  • Abstract : Mimetic Difference Schemes are based on Mimetic Difference Operators which are discrete analogs of the continuous first order invariant operators divergence, gradient and curl. They have been used for quite some time to solve effectively a wide range of partial differential equations. In this mini symposium we will present recent advances on mimetic methods including energy conservation, stability analysis and extension of stability region via Mollification techniques. Numerical examples will be presented to illustrate the effectiveness of the methods
  • Organizer(s) : Jose E. Castillo
  • Classification : 65Nxx
  • Speakers Info :
    • Jose E Castillo (Computational Science Research Center at San Diego State University)
    • Jorge Villamizar (Universidad Industrial de Santander)
    • Julio Cesar Carrillo (Universidad Industrial de Santander)
    • Anand Srinivasan (Computational Science Research Center at San Diego State University)
  • Talks in Minisymposium :
    • [01714] Fourth-order Mimetic Differential Operators Applied to the Convection-Diffusion Equation: A matrix Stability Analysis
      • Author(s) :
        • Jorge VILLAMIZAR (Universidad Industrial de Santander/Universidad de Los Andes)
        • Larry Mendoza (Universidad Central de Venezuela)
        • Giovanni Calderon (Universidad Industrial de Santander)
        • Otilio Rojas (Universidad Central de Venezuela)
        • Jose E Castillo (Computational Science Research Center at San Diego State University)
      • Abstract : The convection-diffusion equation describes physical phenomena where particles or energy are transferred within a physical system due to the processes of diffusion and convection. In this work, we investigate discretization framework based on the mimetic fourth-order finite-difference staggered-grid Castillo-Grone $(CG)$ operators, which has a sextuple of free parameters. We study the dependency of the stability and precision properties of our numerical scheme based on these CG free parameters, and propose parameters that favor both properties. We compare our results with CG parameters previously mentioned in the literature, including those leading to mimetic operators of minimum bandwidth.
    • [01788] Discrete mollification results in mimetic difference schemes applied to the convection-diffusion-reaction equation
      • Author(s) :
        • Julio Cesar Carrillo-Escobar (Professor)
        • Giovanni Ernesto Calderon-Silva (Universidad Industrial de Santander)
      • Abstract : It is usual to have stability restrictions when using either an explicit finite differences or a mimetic differences schemes, proposed by Castillo in 2003, to obtain numerical solutions for the convection-diffusion-reaction equation of an incompressible fluid with a source term. In this work, we analyze the effects, in precision and stability, when applying the discrete mollification proposed by Acosta in 2008 to these schemes in mimetic differences.
    • [01936] Numerical Energy Conservation Mimetic Scheme For The Advection Equation
      • Author(s) :
        • Anand Srinivasan (San Diego State University)
        • Jose E Castillo (Computational Science Research Center at San Diego State University)
      • Abstract : The advection equation $u_t + \nabla \cdot u = 0$ is a hyperbolic partial differential equation that conserves energy. The numerical solution of the advection equation obtained using the traditional finite difference methods often fails to discretely conserve this numerical energy. Mimetic finite difference methods are structure-preserving and are thus well-suited for hyperbolic problems such as the advection equation. The Mimetic methods of Castillo et al discretely mimic the extended Gauss' divergence theorem, and are therefore a faithful discretization of the continuum vector calculus identities. These methods work on a staggered spatial grid and achieve even order of accuracy at the boundaries as well as the interiors of the domain. The temporal discretization obtained from the Leapfrog scheme is staggered in time. The staggered Mimetic-Leapfrog scheme conserves the numerical energy for the advection equation. In this talk, we present the numerical results illustrating the energy-conserving property of the second order Mimetic-Leapfrog scheme. Stability results of the scheme are also presented.
    • [01939] Energy Conservation for Mimetic Scheme for Advection Equation
      • Author(s) :
        • Jose E Castillo (Computational Science Research Center at San Diego State University)
      • Abstract : Mimetic difference schemes are based on discrete analogs of differential operators, gradient, divergence, and curl, that not only preserve their vector calculus identities but also hold discrete counterparts of integral formulas. A proof of the energy conservation property of second-order mimetic difference schemes is presented for the one-dimensional advection PDE. This proof leverages on the discrete analog of the integration by parts mimetic difference property