Multilevel and local time-stepping discontinuous Galerkin methods for magma dynamics

Abstract

Discontinuous Galerkin (DG) method is presented for numerical modeling of melt migration in a chemically reactive and viscously deforming upwelling mantle column at local chemical equilibrium. DG methods for both advection and elliptic equations provide a robust and efficient solution to the problems of melt migration in the asthenospheric upper mantle. Assembling and solving the elliptic equation is the major bottleneck in these computations. To address this issue, adaptive mesh refinement and local time-stepping methods have been proposed to improve the computational wall time. The robustness of DG methods is demonstrated through two benchmark problems by modeling detailed structure of high-porosity dissolution channels and compaction dissolution waves.

A high‐order numerical study of reactive dissolution in an upwelling heterogeneous mantle: 2. Effect of shear deformation

“…However, a constant mantle viscosity isolates one important effect of the interplay between melt‐rock reaction and shear deformation. Preliminary simulations with a melt‐fraction‐dependent shear viscosity suggested by Hirth and Kohlstedt [] but without matrix shear ( ) in the momentum equation show that variable matrix shear viscosity has a negligible effect on the melt channel formation in the absence of shear deformation [ Tirupathi , ]. A natural extension of this study would be to add the effect of melt and/or stress‐dependent viscosity variations and matrix shear in the model formulation.…”

“…The nondimensionalized equations are solved numerically using a high‐order accurate scheme that consists of a fourth‐order Discontinuous Galerkin (DG) method for spatial discretization and a third‐order low‐storage explicit Runge Kutta method for the time integration [ Schiemenz et al ., ; Tirupathi , ; Tirupathi et al ., ]. The numerical scheme is implemented using the finite element software deal.II [ Bangherth et al ., ].…”

“…Compaction within the dunite channel results in melt channel splitting or bifurcation, producing two symmetrically distributed high‐porosity melt channels within the dunite (Figure b). These first‐order features of the high‐porosity melt channel and dunite channel have been confirmed by additional 2‐D and 3‐D numerical simulations conducted in the unstable wave regime, which is computationally more challenging [ Liang et al ., ; Tirupathi , ]. Spatial distributions of the high‐porosity melt and dunite channels are important to the interpretation of geochemical data observed in basalts and mantle peridotites.…”

High order WENO and DG methods for time-dependent convection-dominated PDEs: A brief survey of several recent developments

“…We list below only a few examples over the past year (since 2015). Recent applications of DG methods can be found in the simulations of the Cahn-Hilliard-Brinkman system [85], compressible flow in the transonic axial compressor [190], computational astrophysics [197], computational geosciences [221], elastodynamics [53], flow instabilities [51], Fokker-Planck equations [152], fractional PDEs [111,243], front propagation with obstacles [17], functionalized Cahn-Hilliard equation [87], interfaces [278], magnetohy-drodynamics [265], moment closures for kinetic equations [2], multi-phase flow and phase transition [52,169], Navier-Stokes and Boussinesq equations [64,224], nonlinear Schrodinger equation [86,149,158], ocean waves [192], population models [112], porous media [84], rarefied gas [212], semiconductor device simulation [155], shallow water equations [73], thin film epitaxy problem [247], traffic flow and networks [21], three-dimensional flows [175], turbulent flows [246], underwater explosion [235], viscous surface wave [245], and wavefield modeling [95]. This very incomplete list over just one year period clearly demonstrates the wide-spread application of the DG method in computational science and engineering.…”

A hybridized discontinuous Galerkin method for a linear degenerate elliptic equation arising from two-phase mixtures

“…As such, it is well-suited to problems with large gradients including shocks and with complex geometries, and large-scale simulations demanding parallel implementations. In particular, for numerical modeling of magma dynamics, the DG methods have been used to study the interaction between the fluid melt and the solid matrix [34,31], and to include a porosity-dependent bulk viscosity and a solid upwelling effect [35]. In spite of these advantages, DG methods for steady state and/or time-dependent problems that require implicit time-integrators are more expensive in comparison to other existing numerical methods, since DG typically has many more (coupled) unknowns.…”