MAST solution of advection problems in irrotational flow fields

Abstract A new numerical–analytical Eulerian procedure is proposed for the solution of convection-dominated problems in the case of existing scalar potential of the flow field. The methodology is based on the conservation inside each computational elements of the 0th and 1st order effective spatial moments of the advected variable. This leads to a set of small ODE systems solved sequentially, one element after the other over all the computational domain, according to a MArching in Space and Time technique. The proposed procedure shows the following advantages: (1) it guarantees the local and global mass balance; (2) it is unconditionally stable with respect to the Courant number, (3) the solution in each cell needs information only from the upstream cells and does not require wider and wider stencils as in most of the recently proposed higher-order methods; (4) it provides a monotone solution. Several 1D and 2D numerical test have been performed and results have been compared with analytical solutions, as well as with results provided by other recent numerical methods.

[1]  Chi-Wang Shu,et al.  Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems , 2001, J. Sci. Comput..

[2]  A. Iserles.  Generalized Leapfrog Methods , 1986 .

[3]  E. Toro.  Riemann Solvers and Numerical Methods for Fluid Dynamics , 1997 .

[4]  T. F. Russell,et al.  An Eulerian-Lagrangian localized adjoint method for the advection-diffusion equation , 1990 .

[5]  S. N. Milford,et al.  Eulerian‐Lagrangian Solution of the Convection‐Dispersion Equation in Natural Coordinates , 1984 .

[6]  Gour-Tsyh Yeh,et al.  A Lagrangian‐Eulerian Method with zoomable hidden fine‐mesh approach to solving advection‐dispersion equations , 1990 .

[7]  Richard W. Healy,et al.  A finite‐volume Eulerian‐Lagrangian Localized Adjoint Method for solution of the advection‐dispersion equation , 1993 .

[8]  Tullio Tucciarelli,et al.  An explicit unconditionally stable numerical solution of the advection problem in irrotational flow fields , 2004 .

[9]  Kyōto Daigaku. Sūgakuka.  Lectures in mathematics , 1968 .

[10]  S. Osher,et al.  Uniformly high order accurate essentially non-oscillatory schemes, 111 , 1987 .

[11]  Chi-Wang Shu,et al.  The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case , 1990 .

[12]  A. Baptista,et al.  On the role of tracking on Eulerian-Lagrangian solutions of the transport equation , 1998 .

[13]  B. Després,et al.  Un schéma non linéaire anti-dissipatif pour l'équation d'advection linéaire , 1999 .

[14]  S. Osher,et al.  Some results on uniformly high-order accurate essentially nonoscillatory schemes , 1986 .

[15]  Chi-Wang Shu,et al.  TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems , 1989 .

[16]  Chi-Wang Shu,et al.  The Runge-Kutta Discontinuous Galerkin Method for Conservation Laws V , 1998 .

[17]  R. LeVeque.  Numerical methods for conservation laws , 1990 .

[18]  S. Osher,et al.  Weighted essentially non-oscillatory schemes , 1994 .

[19]  T. F. Russell,et al.  An overview of research on Eulerian-Lagrangian localized adjoint methods (ELLAM). , 2002 .

[20]  Leonardo Noto,et al.  DORA Algorithm for Network Flow Models with Improved Stability and Convergence Properties , 2001 .

[21]  Tullio Tucciarelli,et al.  Finite-element modeling of floodplain flow , 2000 .

[22]  S. Osher,et al.  Uniformly High-Order Accurate Nonoscillatory Schemes. I , 1987 .

[23]  J. Bear,et al.  An adaptive pathline-based particle tracking algorithm for the Eulerian–Lagrangian method , 2000 .

[24]  Chi-Wang Shu,et al.  TVB Runge-Kutta local projection discontinuous galerkin finite element method for conservation laws. II: General framework , 1989 .

[25]  Eleuterio F. Toro,et al.  ADER: Arbitrary High Order Godunov Approach , 2002, J. Sci. Comput..

[26]  P. Woodward,et al.  The Piecewise Parabolic Method (PPM) for Gas Dynamical Simulations , 1984 .

[27]  Chi-Wang Shu,et al.  Efficient Implementation of Weighted ENO Schemes , 1995 .

[28]  P. Colella.  Multidimensional upwind methods for hyperbolic conservation laws , 1990 .

[29]  Steven M. Gorelick,et al.  Semi‐analytical method for departure point determination , 2005 .

[30]  Antonio E. de M Baptista,et al.  Solution of advection-dominated transport by Eulerian-Lagrangian methods using the backwards method of characteristics , 1987 .

[31]  Armin Iske,et al.  ADER schemes on adaptive triangular meshes for scalar conservation laws , 2005 .

[32]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov's method , 1979 .

[33]  Computer evaluation of high order numerical schemes to solve advective transport problems , 1995 .

[34]  K. Ohgushi,et al.  Refined Numerical Scheme for Advective Transport in Diffusion Simulation , 1997 .

[36]  Chi-Wang Shu,et al.  The Runge-Kutta local projection $P^1$-discontinuous-Galerkin finite element method for scalar conservation laws , 1988, ESAIM: Mathematical Modelling and Numerical Analysis.

[37]  Q. Tran,et al.  High-order monotonicity-preserving compact schemes for linear scalar advection on 2-D irregular meshes , 2002 .

[38]  Nick Martin,et al.  MOD_FreeSurf2D: A MATLAB surface fluid flow model for rivers and streams , 2005, Comput. Geosci..