![]() New numerical features are also added, including: (1) The \(\nabla \cdot \mathbf \) constraint error cleaning procedure via an easy-to-use fast multigrid Poisson solver, (2) The Courant-number-insensitive method that reduces the numerical viscosity without generating any instability, (3) The time-integration by multiple time stepping, (4) The time-dependent boundary condition at the subsonic region by limiting the mass flux escaping through the solar surface. 4 is applied to solar wind simulation with this grid system. The SIP-CESE MHD model introduced in Chap. This chapter introduces the implementation of the SIP-CESE MHD model in six component overset grid system with the aim of mitigating the problem of singularity and mesh convergence near the poles. Indeed, the spectral analysis of synthetic models for the internal structure allows us to evaluate the influence of several key parameters of the internal dynamics (viscosity variations, internal heating). A spectral method is described in a dimensionless framework. We thus propose systematic evaluation of geoid and dynamic topography from 3D spherical numerical models of internal dynamics (thermal convection for a variable viscosity fluid). When seismological data are not available, measuring a planet's gravity field and relief provides a first order constraint on the internal structure. The second part of this work deals with internal dynamics of terrestrial planets and its link with geophysical observables. ![]() Mars hypsometry is related to processes that, on Earth, are associated with thermal isostasy and, on Mars, correspond to the location of this primordial ocean. The related variations of the gravity potential and the estimation of the geographical limits of this contact highlight the last Noachian deformation of the planet and allow us to determine the characteristics of the primordial martian ocean. Mars Global Surveyor data are used to study the martian Deuteronilus shoreline. For a grid mesh of a reasonable size (6 × 64 × 64 × 64) a very good agreement (to within ∼1 per cent) is found up to spherical harmonic degree 15. The flow solver is then tested extensively against a precise spectral program, producing response functions for geoid as well as bottom and surface topographies. The evaluation of geoid and surface dynamic topography from the gridded data is performed in the spectral domain. Benchmarks of thermal convection are then presented on steady-state tetrahedral and cubic solutions and time-dependent cases with a good agreement with the few recent programs developed to solve this problem.Ī dimensionless framework is proposed for the calculation of geoid and topography introducing two dimensionless numbers: such a formulation provides a good basis for the systematic study of the geoid and surface dynamic topography associated to the convection calculations. An investigation of various numerical advection schemes is proposed: we opted for a high-resolution, flux-limiter method. The grid mesh is based on the ‘cubed sphere’ technique that divides the shell into six identical blocks. Our tool is based on the simulation of thermal convection with variable viscosity in a spherical shell with a finite-volume formulation. We present a new numerical method to describe the internal dynamics of planetary mantles through the coupling of a dynamic model with the prediction of geoid and surface topography. Under these two AMR implementations, tests are carried out for the solar wind background study, and numerical results are compared with the observations in the solar corona and in interplanetary space from the Solar and Heliospheric Observatory (SOHO) and spacecraft data from OMNI. 5, while the other is implemented for the CESE solver of associated partial differential equations (PDEs) in the reference space of curvilinear coordinates transformed from the original governing partial differential equations in the physical space. Two AMR realization strategies are employed: one uses a solution adaptive technique directly for the CESE solver in the six-component grid system introduced in Chap. This chapter is devoted to the adaptive mesh refinement (AMR) implementation of Solar-Interplanetary space-time conservation element and solution element (CESE) magnetohydrodynamic model (SIP-CESE MHD model) with the aid of the parallel AMR package PARAMESH. The scalable, massively parallel, block-based, adaptive-mesh refinement (AMR) allows resolving such disparate spatial and temporal scales throughout the computational domain with even less cells but can generate a necessary resolution. Coronal-heliospheric space is characterized by disparate temporal and spatial scales as well as by different relevant physics in different domains such as the corona and the inner heliosphere.
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