This work is devoted to the numerical approximation of three-temperature multi-material hydrodynamics. Such systems are subject to stiff phenomena which require specific care during the discretization. In particular, the so-called discrete equation method is here applied to the radiation transport, in the optically-thick limit. This strategy is shown to be accurate in the presence of in-cell interfaces while being simpler than standard interface reconstruction techniques. It is then incorporated into a three-temperature multi-material scheme whose implicit temporal discretization is based on convex combinations. Stiff test cases eventually establish the scheme’s robustness.

Most high-order computational fluid dynamics methods for compressible flows are based on the Riemann solver for the flux evaluation and high-order interpolation or reconstruction such as the Weighted Essential Non-Oscillatory (WENO) scheme for spatial accuracy. The advantage of this kind of combination is the easy implementation and the ability to achieve the required spatial accuracy. However, despite the extensive research on high-order spatial reconstruction in the past, solvers coupling high-order space and time schemes have not been systematically evaluated. In this paper, based on the same fifth-order finite volume method (FVM), comparisons of the performance of the same flux solver with different reconstructions and the same reconstruction but different flux solvers are carried out on a structured mesh. For reconstruction, the TENO scheme and classic WENO-Z reconstruction have been chosen as representative methods. Meanwhile, for the flux solver, Lax-Friedrichs (LF) Riemann solver, HLLC solver, and GKS are considered. Through a series of simulated comparison cases, the unique characteristics of GKS and TENO have been demonstrated. Overall, the comparisons suggest that proper spatial and temporal coupling is important for accurate shock and vortex capturing.

We present a graph-based numerical method for solving hyperbolic systems of conservation laws using discontinuous finite elements. This work fills important gaps in the theory as well as practice of graph-based schemes. In particular, four building blocks required for the implementation of flux-limited graph-based methods are developed and tested: a first-order method with mathematical guarantees of robustness; a high-order method based on the entropy viscosity technique; a procedure to compute local bounds; and a convex limiting scheme. Two important features of the current work are the fact that (i) boundary conditions are incorporated into the mathematical theory as well as the implementation of the scheme. For instance, the first-order version of the scheme satisfies pointwise entropy inequalities including boundary effects for any boundary data that is admissible; (ii) sub-cell limiting is built into the convex limiting framework. This is in contrast to the majority of the existing methodologies that consider a single limiter per cell providing no sub-cell limiting capabilities. From a practical point of view, the implementation of graph-based methods is algebraic, meaning that they operate directly on the stencil of the spatial discretization. In principle, these methods do not need to use or invoke loops on cells or faces of the mesh. Finally, we verify convergence rates on various well-known test problems with differing regularity. We propose a simple test in order to verify the implementation of boundary conditions and their convergence rates.

In this paper, we develop a general framework for the design of the arbitrary high-order well-balanced discontinuous Galerkin (DG) method for hyperbolic balance laws, including the compressible Euler equations with gravitation and the shallow water equations with horizontal temperature gradients (referred to as the Ripa model). Not only does the hydrostatic equilibrium include the more complicated isobaric steady state in the Ripa system, but our scheme is also well-balanced for the exact preservation of the moving equilibrium state. The strategy adopted is to approximate the equilibrium variables in the DG piecewise polynomial space, rather than the conservative variables, which is pivotal in the well-balanced property. Our approach provides flexibility in combination with any consistent numerical flux, and it is free of the reference equilibrium state recovery and the special source term treatment. This approach enables the construction of a well-balanced method for non-hydrostatic equilibria in Euler equations. Extensive numerical examples such as moving or isobaric equilibria validate the high order accuracy and exact equilibrium preservation for various flows given by hyperbolic balance laws. With a relatively coarse mesh, it is also possible to capture small perturbations at or close to steady flow without numerical oscillations.

We develop new low-dissipation central-upwind (LDCU) schemes for nonlinear elasticity equations in heterogeneous media. In general, central-upwind schemes belong to the class of finite-volume Godunov-type schemes, which consist of three steps: reconstruction, evolution, and projection onto the original grid. In our new method, the evolution is performed in the standard way by integrating the system over the space-time control volumes. However, the reconstruction and projection are performed in a special manner. First, we take into account the fact that the conservative variables (strain and momentum) are discontinuous across the material interfaces, while the flux variables (velocity and strain) are continuous there: we therefore reconstruct the flux variables. Second, we use a special projection recently introduced in [A. Kurganov and R. Xin, J. Sci. Comput., 96, 2023] to complete the derivation of the LDCU scheme. Our numerical experiments demonstrate that the developed schemes are capable of accurately resolving waves with dispersive behavior that over a long period of time evolve into solitary waves.

This paper proposes a Kolmogorov high order deep neural network (K-HOrderDNN) for solving high-dimensional partial differential equations (PDEs), which improves the high order deep neural networks (HOrderDNNs). HOrderDNNs have been demonstrated to outperform conventional DNNs for high frequency problems by introducing a nonlinear transformation layer consisting of $(p+1)^d$ basis functions. However, the number of basis functions grows exponentially with the dimension $d,$ which results in the curse of dimensionality (CoD). Inspired by the Kolmogorov Superposition Theorem (KST), which expresses a multivariate function as superpositions of univariate functions and addition, K-HOrderDNN utilizes a HOrderDNN to efficiently approximate univariate inner functions instead of directly approximating the multivariate function, reducing the number of introduced basis functions to $d(p+1).$ We theoretically demonstrate that CoD is mitigated when target functions belong to a dense subset of continuous multivariate functions. Extensive numerical experiments show that: for high-dimensional problems $(d=10, 20, 50)$ where HOrderDNNs$(p>1)$ are intractable, K-HOrderDNNs$(p>1)$ exhibit remarkable performance. Specifically, when $d=10,$ K-HOrderDNN$(p=7)$ achieves an error of $4.40E−03,$ two orders of magnitude lower than that of HOrderDNN$(p=1)$ (see Table 10); for high frequency problems, K-HOrderDNNs$(p>1)$ can achieve higher accuracy with fewer parameters and faster convergence rates compared to HOrderDNNs (see Table 8).

Geothermal energy is a sustainable baseload source recognized for its ability to provide clean energy on a large scale. Advanced Geothermal Systems (AGS)—offer promising prototypes—employ a closed-loop vascular layout that runs deep beneath the Earth’s surface. A working fluid (e.g., water or supercritical carbon dioxide (sCO2)) circulates through the vasculature, entering the subsurface at the inlet and exiting at the outlet with an elevated temperature. For designing and performing cost-benefit analysis before deploying large-scale projects and maintaining efficiency while enabling real-time monitoring during the operational phase, modeling offers cost-effective solutions; often, it is the only available option for performance assessment. A knowledge gap exists due to the lack of a fast predictive modeling framework that considers the vascular intricacies, particularly the jumps in the solution fields across the channel. Noting that the channel diameter is considerably smaller in scale compared to the surrounding geological domain, we develop a reduced-order modeling (ROM) framework for closed-loop geothermal systems. This ROM incorporates the jump conditions and provides a quick and accurate prediction of the temperature field, including the outlet temperature, which directly correlates with the power production capacity and thermal draw-down. We demonstrate the predictive capabilities of the framework by establishing the uniqueness of the solutions and reporting representative numerical solutions. The modeling framework and the predictions reported in this paper benefit the closed-loop geothermal community, enabling them to determine the system’s performance and optimal capacity.

Non-overlapping domain decomposition methods are well-suited for addressing interface problems across various disciplines, where traditional numerical simulations often require the use of interface-fitted meshes or technically designed basis functions. To remove the burden of mesh generation and to effectively tackle with the flux transmission condition, a novel mesh-free scheme, i.e., the Dirichlet-Neumann learning algorithm, is studied in this work for solving the benchmark elliptic interface problems with high-contrast coefficients and irregular interfaces. By resorting to the variational principle, we carry out a rigorous error analysis to evaluate the discrepancy caused by the boundary penalty treatment for each decomposed subproblem, which paves the way for realizing the Dirichlet-Neumann algorithm using neural network extension operators. Through experimental validation on a series of testing problems in two and three dimensions, our methods demonstrate superior performance over other alternatives even in scenarios with inaccurate flux predictions at the interface.

In this paper, we develop a novel control volume method that is locally conservative and locking-free for linear elasticity problem on quadrilateral grids. The symmetry of stress is weakly imposed through the introduction of a Lagrange multiplier. As such, the method involves three unknowns: stress, displacement and rotation. To ensure the well-posedness of the scheme, a pair of carefully defined finite element spaces is used for the stress, displacement and rotation such that the inf-sup condition holds. An appealing feature of the method is that piecewise constant functions are used for the approximations of stress, displacement and rotation, which greatly simplifies the implementation. In particular, the stress space is defined delicately such that the stress bilinear form is localized around each vertex, which allows for the local elimination of the stress, resulting in a cell-centered system. By choosing different definitions of the space for rotation, we develop two variants of the method. In particular, the first method uses a constant function for rotation over the interaction region, which allows for further elimination and results in a cell-centered system involving displacement only. A rigorous error analysis is performed for the proposed scheme. We show the optimal convergence for $L^2$-error of the stress and rotation. Moreover, we can also prove the superconvergence for $L^2$-error of displacement. Extensive numerical simulations indicate that our method is efficient and accurate, and can handle problems with discontinuous coefficients.