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This paper proposes and analyzes two fully discrete mixed interior penalty discontinuous Galerkin (DG) methods for the fourth order nonlinear Cahn-Hilliard equation. Both methods use the backward Euler method for time discretization and interior penalty DG methods for spatial discretization. They differ from each other on how the nonlinear term is treated; one of them is based on fully implicit time-stepping and the other uses energy-splitting time-stepping. The primary goal of the paper is to prove the convergence of the numerical interfaces of the DG methods to the interface of the Hele-Shaw flow. This is achieved by establishing error estimates that depend on ε⁻¹ only in some low polynomial orders, instead of exponential orders. Similar to [X. Feng and A. Prohl, Numer. Math., 74 (2004), pp. 47-84], the crux is to prove a discrete spectrum estimate in the discontinuous Galerkin finite element space. However, the validity of such a result is not obvious because the DG space is not a subspace of the (energy) space H¹(Ω) and it is larger than the finite element space. This difficulty is overcome by a delicate perturbation argument which relies on the discrete spectrum estimate in the finite element space proved by Feng and Prohl. Numerical experiment results are also presented to gauge the theoretical results and the performance of the proposed fully discrete mixed DG methods.

Daqing Oilfield is rich in low permeability reservoirs. In order to fully understand the development dynamics of ultra-low permeability reservoirs, this paper takes Daqing Oilfield as an example and studies the productivity dynamics of fractured horizontal Wells by numerical simulation method. The results show that the productivity of fractured horizontal Wells decreases exponentially with the decrease of formation permeability. The productivity of fractured horizontal well increases with fracture length and decreases with fracture spacing. With the increase of formation coefficient and reservoir thickness, the productivity increase of fractured horizontal Wells decreases. In the case of small well spacing, the smaller the formation coefficient and reservoir thickness, the higher the productivity of fractured horizontal Wells. Compared with conventional vertical Wells, fractured horizontal Wells have better production performance.

In various engineering projects such as mineral extraction, hydropower resource utilization, railway construction, and geological hazard mitigation, rock engineering is often encountered. Furthermore, dynamic loads and moisture content exert notable influence on the energy transformation processes within rocks. Yet, the specific interplay of dynamic loading and water's impact on the energy conversion mechanism within the sandstone remains unexplored. To address this gap, this study conducted impact loading experiments on sandstone, elucidating the rock’s mechanical response under these conditions and unraveling the underlying energy conversion mechanisms. It was observed that the strength of sandstone exhibits a direct correlation with impact velocity. Moreover, employing energy calculation principles, we established a connection between moisture content and the sandstone’s internal energy conversion properties. The study also delved into the microscopic fracture mechanisms within the sandstone, ultimately concluding that both water content and dynamic loading have a significant impact on these microscopic fracture mechanisms.

In construction engineering, rock is an important building material. During the construction process, layered rock masses are typically subjected to varying dynamic load disturbances under triaxial loads. It is thus essential to investigate the mechanical response of layered rocks under various disturbances of the triaxial loads. By using a three-dimensional SHPB, triaxial dynamic compression tests with various impact dynamic load disturbances and identical triaxial static loads were carried out on sandstones with differing bedding angles. The impact pressures were 0.8, 1.2, and 1.6 MPa, and the bedding angles were 0°, 30°, 45°, 60°, and 90°. The results showed that the ductility of the sandstone considerably increased under triaxial static loading. With the increasing bedding angle, the sandstone’s dynamic strength and coupling strength first declined and subsequently rose. As the impact pressure increased, the reflective energy ratio, peak strain, and dynamic growth factor of the sandstone essentially rose progressively. The bedding angles and dynamic loads had a major impact on the damage pattern of the layered sandstones. Additionally, a constitutive model considering bedding angle, dynamic load, and static load was established and verified. The constitutive model was able to accurately characterize the dynamic behavior of the rock under load disturbances.

The shallow water equations model flows in rivers and coastal areas and have wide applications in ocean, hydraulic engineering, and atmospheric modeling. In “Xing et al. Adv. Water Resourc. 33: 1476–1493, 2010 )”, the authors constructed high order discontinuous Galerkin methods for the shallow water equations which can maintain the still water steady state exactly, and at the same time can preserve the non-negativity of the water height without loss of mass conservation. In this paper, we explore the extension of these methods on unstructured triangular meshes. The simple positivity-preserving limiter is reformulated, and we prove that the resulting scheme guarantees the positivity of the water depth. Extensive numerical examples are provided to verify the positivity-preserving property, well-balanced property, high-order accuracy, and good resolution for smooth and discontinuous solutions.

Generalized Gauss–Radau (GGR) projections are global projection operators that are widely used for the error analysis of discontinuous Galerkin (DG) methods with generalized numerical fluxes. In previous work, GGR projections were constructed for Cartesian meshes and analyzed through an algebraic approach. In this paper, we first present an alternative energy approach for analyzing the one-dimensional GGR projection, which does not require assembling and explicitly solving a global system over the entire computational domain as that in the algebraic approach. We then generalize this energy argument to construct a global projection operator on special simplex meshes in multidimensions satisfying the so-called flow condition. With this projection, optimal error estimates are proved for upwind-biased DG methods for the linear advection equation on these meshes, which generalizes the error analysis for the purely upwind case by Cockburn et al. (SIAM J Numer Anal 46(3):1250–1265, 2008) in a time-dependent setting.

Superparameterization (SP) is a large-scale modeling system with explicit representation of small-scale and mesoscale processes provided by a cloud-resolving model (CRM) embedded in each column of a large-scale model. New efficient sparse space–time algorithms based on the original idea of SP are presented. The large-scale dynamics are unchanged, but the small-scale model is solved in a reduced spatially periodic domain to save the computation cost following a similar idea applied by one of the authors for aquaplanet simulations. In addition, the time interval of integration of the small-scale model is reduced systematically for the same purpose, which results in a different coupling mechanism between the small- and large-scale models. The new algorithms have been applied to a stringent two-dimensional test suite involving moist convection interacting with shear with regimes ranging from strong free and forced squall lines to dying scattered convection as the shear strength varies. The numerical results are compared with the CRM and original SP. It is shown here that for all of the regimes of propagation and dying scattered convection, the large-scale variables such as horizontal velocity and specific humidity are captured in a statistically accurate way (pattern correlations above 0.75) based on space–time reduction of the small-scale models by a factor of ⅓; thus, the new efficient algorithms for SP result in a gain of roughly a factor of 10 in efficiency while retaining a statistical accuracy on the large-scale variables. Even the models with ⅙ reduction in space–time with a gain of 36 in efficiency are able to distinguish between propagating squall lines and dying scattered convection with a pattern correlation above 0.6 for horizontal velocity and specific humidity. These encouraging results suggest the possibility of using these efficient new algorithms for limited-area mesoscale ensemble forecasting.

Shallow water equations with horizontal temperature gradients, also known as the Ripa system, are used to model flows when the temperature fluctuations play an important role. These equations admit steady state solutions where the fluxes and source terms balance each other. We present well-balanced discontinuous Galerkin methods for the Ripa model which can preserve the still-water or the general moving-water equilibria. The key ideas are the recovery of well-balanced states, separation of the solution into the equilibrium and fluctuation components, and appropriate approximations of the numerical fluxes and source terms. The same framework is also extended to design well-balanced methods for the constant height and isobaric steady state solutions of the Ripa model. Numerical examples are presented to verify the well-balanced property, high order accuracy, and good resolution for both smooth and discontinuous solutions.

Determining the finite-amplitude preconditioned states in the hurricane embryo, which lead to tropical cyclogenesis, is a central issue in contemporary meteorology. In the embryo there is competition between different preconditioning mechanisms involving hydrodynamics and moist thermodynamics, which can lead to cyclogenesis. Here systematic asymptotic methods from applied mathematics are utilized to develop new simplified moist multi-scale models starting from the moist anelastic equations. Three interesting multi-scale models emerge in the analysis. The balanced mesoscale vortex (BMV) dynamics and the microscale balanced hot tower (BHT) dynamics involve simplified balanced equations without gravity waves for vertical vorticity amplification due to moist heat sources and incorporate nonlinear advective fluxes across scales. The BMV model is the central one for tropical cyclogenesis in the embryo. The moist mesoscale wave (MMW) dynamics involves simplified equations for mesoscale moisture fluctuations, as well as linear hydrostatic waves driven by heat sources from moisture and eddy flux divergences. A simplified cloud physics model for deep convection is introduced here and used to study moist axisymmetric plumes in the BHT model. A simple application in periodic geometry involving the effects of mesoscale vertical shear and moist microscale hot towers on vortex amplification is developed here to illustrate features of the coupled multi-scale models. These results illustrate the use of these models in isolating key mechanisms in the embryo in a simplified content.

Euler equations under gravitational field admit hydrostatic equilibrium state where the flux produced by the pressure is exactly balanced by the gravitational source term. In this paper, we present well-balanced Runge–Kutta discontinuous Galerkin methods which can preserve the isothermal hydrostatic balance state exactly and maintain genuine high order accuracy for general solutions. To obtain the well-balanced property, we first reformulate the source term, and then approximate it in a way which mimics the discontinuous Galerkin approximation of the flux term. Extensive one- and two-dimensional simulations are performed to verify the properties of these schemes such as the exact preservation of the hydrostatic balance state, the ability to capture small perturbation of such state, and the genuine high order accuracy in smooth regions.