Upstream Nonoscillatory Advection Schemes

Abstract Upstream nonoscillatory (UNO) advection schemes are derived by optimizing existing classical advection schemes and combining them in different monotonic zones to avoid flux limiters for simplicity. The UNO schemes are extended to irregular grids in the form of upstream midflux linear interpolation with symmetrical gradients and are adapted to multidimensions with an advective–conservative operator. They are given in finite-volume flux form and hence are consistent and conservative. They also preserve constancy and linear correlation. Implicit numerical diffusivity of these schemes is also derived and used as a guideline for the selection of advection schemes. One- and two-dimesional tests are used for comparisons with their classical counterparts. Multiple-cell grids are used to test the irregular grid formulation and demonstrate their performance. The simple second-order UNO2 scheme may be accurate enough when the physical diffusion or numerical smoothing term is larger than the numerical diffus...

[1]  M. Berger,et al.  Adaptive mesh refinement for hyperbolic partial differential equations , 1982 .

[2]  Ming Xue,et al.  High-Order Monotonic Numerical Diffusion and Smoothing , 2000 .

[3]  B. P. Leonard,et al.  Conservative Explicit Unrestricted-Time-Step Multidimensional Constancy-Preserving Advection Schemes , 1996 .

[4]  P. Rasch.  Conservative Shape-Preserving Two-Dimensional Transport on a Spherical Reduced Grid , 1994 .

[5]  P. Sweby.  High Resolution Schemes Using Flux Limiters for Hyperbolic Conservation Laws , 1984 .

[6]  P. Lax,et al.  Systems of conservation laws , 1960 .

[7]  Jason C. Knievel,et al.  Explicit Numerical Diffusion in the WRF Model , 2007 .

[8]  P. Roe.  Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes , 1997 .

[9]  J. Fromm.  A method for reducing dispersion in convective difference schemes , 1968 .

[10]  Nigel Wood,et al.  A monotonic and positive–definite filter for a Semi‐Lagrangian Inherently Conserving and Efficient (SLICE) scheme , 2005 .

[11]  C. Walcek,et al.  A simple but accurate mass conservative, peak-preserving, mixing ratio bounded advection algorithm with FORTRAN code , 1998 .

[12]  Shian‐Jiann Lin,et al.  Multidimensional Flux-Form Semi-Lagrangian Transport Schemes , 1996 .

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

[14]  Frédéric Hourdin,et al.  The Use of Finite-Volume Methods for Atmospheric Advection of Trace Species. Part I: Test of Various Formulations in a General Circulation Model , 1999 .

[15]  Lawrence L. Takacs,et al.  A Two-Step Scheme for the Advection Equation with Minimized Dissipation and Dispersion Errors , 1985 .

[16]  B. P. Leonard,et al.  The ULTIMATE conservative difference scheme applied to unsteady one-dimensional advection , 1991 .

[17]  H. Tolman,et al.  Development and Implementation of Wind-Generated Ocean Surface Wave Modelsat NCEP , 2002 .

[18]  F. Schimmel,et al.  Use of (weighted) essentially non‐oscillatory advection schemes in a mesoscale model , 2006 .

[19]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme. IV. A new approach to numerical convection , 1977 .

[20]  L. Margolin,et al.  A rationale for implicit turbulence modelling , 2001 .

[21]  David L. Book,et al.  Flux-corrected transport II: Generalizations of the method , 1975 .

[22]  M. Prather.  Numerical advection by conservation of second-order moments. [for trace element spatial distribution and chemical interaction in atmosphere] , 1986 .

[23]  P. Smolarkiewicz.  A Fully Multidimensional Positive Definite Advection Transport Algorithm with Small Implicit Diffusion , 1984 .

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

[25]  W. Grabowski,et al.  The multidimensional positive definite advection transport algorithm: nonoscillatory option , 1990 .

[26]  A. Staniforth,et al.  A new dynamical core for the Met Office's global and regional modelling of the atmosphere , 2005 .

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

[28]  E. Holm,et al.  A Fully Two-Dimensional, Nonoscillatory Advection Scheme for Momentum and Scalar Transport Equations , 1995 .

[29]  S. Zalesak.  Fully multidimensional flux-corrected transport algorithms for fluids , 1979 .

[30]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme. II. Monotonicity and conservation combined in a second-order scheme , 1974 .

[31]  M. Maqueda,et al.  Performance of a second‐order moments advection scheme in an Ocean General Circulation Model , 2006 .

[32]  Richard B. Rood,et al.  Application of a Monotonic Upstream-biased Transport Scheme to Three-Dimensional Constituent Transport Calculations , 1991 .

[33]  A. Bott.  Monotone Flux Limitation in the Area-preserving Flux-form Advection Algorithm , 1992 .

[34]  A Multiple-Cell Flat-Level Model for Atmospheric Tracer Dispersion over Complex Terrain , 2003 .

[35]  A. Staniforth,et al.  Comments on “Swolarkiewicz's Deformational Flow” , 1987 .

[36]  W. P. Crowley,et al.  NUMERICAL ADVECTION EXPERIMENTS1 , 1968 .

[37]  Jay P. Boris,et al.  Flux-corrected transport. I. SHASTA, a fluid transport algorithm that works , 1973 .

[38]  Joseph M. Prusa,et al.  Dynamic Grid Adaptation Using the MPDATA Scheme , 2002 .

[39]  P. Smolarkiewicz.  The Multi-Dimensional Crowley Advection Scheme , 1982 .

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

[41]  J. Thuburn.  TVD Schemes, Positive Schemes, and the Universal Limiter , 1997 .

[42]  James C. McWilliams,et al.  Quasi-Monotone Advection Schemes Based on Explicit Locally Adaptive Dissipation , 1998 .

[43]  B. W. Atkinson,et al.  An inert tracer dispersion scheme for use in a mesoscale atmospheric model , 2000 .

[44]  J. Thuburn.  Multidimensional Flux-Limited Advection Schemes , 1996 .

[45]  B. P. Leonard,et al.  A stable and accurate convective modelling procedure based on quadratic upstream interpolation , 1990 .

[46]  R. Pielke,et al.  The forward-in-time upstream advection scheme:extension to higher orders , 1987 .

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

[48]  Andreas Bott.  The Monotone Area-preserving Flux-Form Advection Algorithm: Reducing the Time-splitting Error in Two-Dimensional Flow Fields , 1993 .

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

[50]  I. Bohachevsky,et al.  Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics , 1959 .

[51]  A. J. Gadd.  A numerical advection scheme with small phase speed errors , 1978 .