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Session Overview
MS 17: High Oder Mimetic Differences and Applications
Wednesday, 14/July/2021:
4:10pm - 6:10pm

Session Chair: Jose E Castillo
Virtual location: Zoom 2

Session Abstract

Mimetic differences operators have been used more and more frequently to construct

numerical schemes for solving partial differential equations with variable degree of success.

There are many researches currently active in this area pursuing different approaches to

achieve this goal and many algorithms have been developed along these lines. Loosely

speaking, "mimetic methods" have discrete structures that mimic vector calculus identities and

theorems. These make the numerical schemes based on mimetic difference operators more

faithful to the physics of the problem under investigation. Specific approaches to discretization

have achieved this compatibility following different paths, and with diverse degrees of

generality in relation to the problems solved and the order of accuracy obtainable. In this

session advances of High Order Mimetic differences methods and applications will be


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4:10pm - 4:40pm

Symmetry-preserving finite-difference discretizations of arbitrary order on structured curvilinear staggered grids

Bas van 't Hof, Mathea J. Vuik


Symmetry-preserving (mimetic) discretization aims to preserve certain properties of a continuous differential operator in its discrete counterpart. For these discretizations, stability and (discrete) conservation of mass, momentum and energy are proven in the same way as for the original continuous model.

In this presentation, we present a new finite-difference symmetry-preserving space discretization. Specifically, the new discretization combines three important requirements:

• it is made for arbitrary orders of accuracy;

• it works for orthogonal and non-orthogonal structured curvilinear staggered meshes;

• it can be applied to a wide variety of continuous operators, including chain rules and advection operators.

The new discretization is applied to several problems including the shallow-water equations on uniform and curvilinear meshes of different resolutions and using different discretization orders. We show exact conservation of discrete mass, momentum and energy and convergence corresponding to expected order.

4:40pm - 5:10pm

Analysis of high order mimetic discretizations on wave propagation and convection–diffusion problems

Jorge VILLAMIZAR1,2, Larry MENDOZA3, Giovanni CALDERÓN2, Otilio ROJAS3,4, José CASTILLO5

1Facultad de Ingeniería, Universidad de Los Andes, ULA , Venezuela; 2Escuela de Matemáticas, Universidad Industrial de Santander, UIS, Colombia; 3Universidad Central de Venezuela, UCV, Venezuela; 4Barcelona Supercomputing Center, Spain; 5Computational Science Research Center, San Diego State University, SDSU, USA

Finite difference (FD) methods used in wave propagation simulations can cause dispersion errors due to the discrete sampling of propagation wavelengths applied by differentiation operators. To minimize such errors, the use of high-order operators allows increasing such sampling and improving numerical accuracy, without densifying numerical meshes. Alternatively, 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. For both physical problems, we here employ a common discretization framework based on the mimetic fourth-order staggered-grid FD Castillo-Grone (CG) operators, that offer a sextuple of free parameters. Special attention has been paid to the dependency of the stability and precision properties of our numerical schemes on these CG parameters. As a reference, our analyses also present results based on CG parameters leading to mimetic operators of minimum bandwidth, that have been previously applied in similar physical problems.

5:10pm - 5:40pm

High Order Mimetic Methods on Curvilinear Grids

Angel Armando Boada Velazco, Johnny Corbino, Jose Castillo

San Diego State University, United States of America

Mimetic differences (MD) are discrete analogs of the differential operators used to describe continuum problems, with successful implementations in the fields of fluid and solid mechanics. These discrete operators satisfy the vector calculus identities of their continuum counterparts, so they are more faithful to the physics. Recent developments have focused on employing MD to solve challenging problems by targeting partial differential equations (PDEs) with rough coefficients, jump discontinuities, and highly nonlinear problems. However, the use of MD on complex geometries has not been studied in detail. An approach that has been used for non-trivial geometries is to work on curvilinear coordinates. In this approach, the governing equations are transformed to curvilinear coordinates, and the domain is mapped to a unit square or volume in 3D. In this work, we explore the viability of solving PDEs defined on non-trivial geometries by way of high-order MD curvilinear operators. In particular, we investigate the discrete conservation of mass, momentum, and energy for some wave models for complex physical space domains. Numerical results will be presented.

5:40pm - 6:10pm

Relaxation Runge Kutta4 (RRK4) and Mimetic Differences for Wave Equations

Anand Srinivasan1, David Ketcheson2, Jose Castillo1

1San Diego State University, United States of America; 2King Abdullah University of Science and Technology

Mimetic discretization methods provide a discrete analog of vector calculus by constructing discrete analogs of the divergence, gradient and curl operators. These operators have been used to develop numerical schemes which have been used in many applications very effectively. For time dependent problems, different time discretization has been used for wave problems. We investigate the combination of fourth order mimetic operators with the recent developed RRK4 by one the co-authors in the paper and its application to wave problems. We investigate stability as well as energy conservation and present examples that demonstrate the effectiveness of our scheme.

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