Numerical dissipation switch for two-dimensional central-upwind schemes

We propose a numerical dissipation switch, which helps to control the amount of numerical dissipation present in central-upwind schemes. Our main goal is to reduce the numerical dissipation without risking oscillations. This goal is achieved with the help of a more accurate estimate of the local propagation speeds in the parts of the computational domain, which are near contact discontinuities and shears. To this end, we introduce a switch parameter, which depends on the distributions of energy in the x- and y-directions. The resulting new central-upwind is tested on a number of numerical examples, which demonstrate the superiority of the proposed method over the original central-upwind scheme.

Implementation of higher-order advection schemes in a numerical ice-sheet model for ice-core studies

<p>Computation of temperature and age fields by numerical ice-sheet models is an important issue for ice-core related studies.  Generally the evolution of temperature and/or age in an ice-sheet model is formulated using an advection equation.  There are many variation of the formulation, which differ in numerical aspects such as stability, accuracy, numerical diffusivity, conservation and/or computational costs.  Saito et al (2020, GMD) implement Rational Constrained Interpolation Profile (RCIP) scheme on vertical 1-d age computation of ice sheet, and demonstrate its efficiency, in particular, to preserve surface mass balance properties recorded at the deposit in terms of annual layer thickness.  Successively, we have been extending the development using RCIP or similar higher-order advection schemes on 3-d age or temperature computation.  In this study, we demonstrate 1-d temperature computation by various numerical schemes including classical upwind schemes and compare the accuracy of those schemes.</p>

Implementation of the RCIP scheme and its performance for 1-D age computations in ice-sheet models

Ice-sheet age computations are formulated using an Eulerian advection equation, and there are many schemes that can be used to solve them numerically. Typically, these differ in numerical characteristics such as stability, accuracy, and diffusivity. Furthermore, although various methods have been presented for ice-sheet age computations, the constrained interpolation profile method and its variants have not been examined in this context. The present study introduces one of its variants, a rational function-based constrained interpolation profile (RCIP) scheme, to one-dimensional ice age computation and demonstrates its performance levels via comparisons with those obtained from first- and second-order upwind schemes. Our results show that the RCIP scheme preserves the pattern of input surface mass balance histories in terms of the vertical profile of internal annual layer thickness better than the other schemes.

MPDATA method for non–uniform mesh

<p> </p><p><strong>MPDATA method for non–uniform mesh</strong></p><p> </p><p>Xinpeng Yuan</p><p>State Key Laboratory of Severe Weather, Chinese Academy of Meteorological Sciences,China </p><p>Meteorological Administration, Beijing 100081, China </p><p> </p><p><strong>Keyword:</strong> Atmospheric dynamics, MPDATA, non–uniform mesh, precision</p><p><strong>Abstract: </strong>MPDATA[1,2]（multidimensional positive definite advection transport algorithm） is proposed by Piotr K. Smolarkiewicz in 1983. This method is used to efficiently solve the advection transport problem of non-negative thermodynamic variables (such as liquid water or water vapor) in the atmospheric dynamics model. This method has been proved to be an effective numerical solution to the advection transport problem for uniform meshes. However, since there is no uniform mesh division on the sphere, the traditional MPDATA method is faced with the incompatibility problem for the non-uniform and quasi-uniform meshing of the sphere, resulting in the numerical algorithm failing to reach the designed second-order accuracy. Firstly, this paper analyzes the insufficiency of traditional MPDATA methods for non-uniform grids. That is, the incompatibility of the first-order numerical scheme and the approximation of boundary derivative.Then the MPDATA method suitable for non-uniform grid is proposed. According to the characteristics of non-uniform grid and the characteristics of well-balance[3] central grid point algorithm, the MPDATA method suitable for 1-d and 2-d complex grid structure is designed. The consistency and positivity of the algorithm are proved by mathematical analysis. Finally, the theoretical proof is verified by numerical simulation.</p><p><strong> </strong></p><p><strong>Reference</strong></p><p>[1] Smolarkiewicz P. A Simple Positive Definite Advection Scheme with Small Implicit Diffusion[J]. Monthly Weather Review. 1983.</p><p>[2] Smolarkiewicz P K, Szmelter J. MPDATA: An edge-based unstructured-grid formulation[J]. Journal of Computational Physics. 2005, 206(2): 624-649.</p><p>[3] Kurganov A, Levy D. Central-Upwind Schemes for the Saint-Venant System[J]. ESAIM: Mathematical Modelling and Numerical Analysis. 2002, 36(3): 397-425.</p><p> </p>

Implementation of RCIP scheme and its performance for 1D age computations in ice-sheet models

Ice sheet age computations are formulated using an Eulerian advection equation, and there are many schemes that can be used to solve them numerically. Typically, these differ in numerical characteristics such as stability, accuracy, and diffusivity. Furthermore, although various methods have been presented for ice sheet age computations, the constrained interpolation profile method and its variants have not been examined in this context. The present study introduces one of its variants, a rational function-based constrained interpolation profile scheme (RCIP) to one-dimensional ice age computation, and demonstrates its performance levels via comparisons with those obtained from first- and second-order upwind schemes. Our results show that the RCIP scheme preserves the pattern of input surface mass balance histories, in terms of the vertical profile of internal annual layer thickness, better than the other schemes.