Abstract. We present a conservation error analysis of non-conservative modified ghost fluid methods (MGFM) for axisymmetric compressible multi-medium flows within a discontinuous Galerkin framework. We propose a corresponding correction strategy. While multi-medium Riemann-solver-based MGFM-type methods (e.g., MGFM, MGFM) excel at simulating strong shock-interface interactions, their inherent non-conservatism can induce mass loss and pressure misalignment in long-time, high-energy-density computations. To address this issue, we theoretically analyze the conservation error introduced by the interface treatment, the cell-wise conservation error and the flux mismatch error. We prove that for MGFM-type methods, the global conservation error CE(M,N) is O(Δt). These errors are then conservatively redistributed to the grid cells adjacent to the interface, thereby eliminating the numerical losses of mass, momentum, and energy. The cell-wise error is redistributed using a local smoothing technique, while the flux mismatch error is allocated according to the wave propagation direction determined by the multi-material Riemann problem. The resulting algorithm integrates high-order discontinuous Galerkin discretizations with one-sided Riemann problems for the symmetry axis, forming a robust numerical framework for axisymmetric multi-medium flows in high-energy-density environments. Numerical experiments demonstrate that the proposed method not only achieves global conservation to machine precision but also effectively eliminates the pressure misalignment at the interface. In long-term, large-scale multi-medium simulations, the predicted key physical quantities are in excellent agreement with experimental data and reference solutions.

We study the Riemann problem for the compressible Euler equations with a stationary coupling interface across which a discontinuity in the heat flux is prescribed. This coupling gives rise to non-conservative effects and models heat addition mechanisms such as condensation-induced waves. Without imposing restrictions on sonic states, we analyze the problem in all Mach number regimes. Lax weak entropy solutions are constructed via half-Riemann problems, and we show that non-uniqueness occurs for a large class of initial data. To address this, we introduce an admissibility criterion derived from the evolutionarity criterion, and we characterize the full structure of admissible Riemann solutions. Our analysis establishes local existence of admissible Riemann solutions provided the heat flux jump is sufficiently small, while also identifying families of initial data for which admissible Riemann solutions cannot exist for any fixed, nonzero heat flux jump. Numerical experiments are included to illustrate the theoretical findings.

In this work, we present a novel path-conservative discontinuous Galerkin (PCDG) method for hypo-elastic plastic solids. While existing PCDG methods are typically based on linear paths—a mature and well-developed approach for fluids—our numerical experiments indicate that this choice leads to unphysical oscillations when applied to the hypo-elastic plastic solid model. Consequently, we propose constructing the numerical fluxes for the non-conservative terms based on a path derived from the HLLC Riemann solver, thereby establishing a core PCDG formulation. Furthermore, to accurately handle the elastic-plastic transition, we implement a radial return mapping method at the Gauss points based on the von Mises yield criterion, subsequently projecting the modified stresses back onto the degrees of freedom. One-dimensional numerical results demonstrate that the proposed HLLC Riemann-solver-based path not only effectively eliminates the unphysical numerical oscillations but also achieves superior accuracy and stable convergence rates. Finally, a two-dimensional Taylor bar impact benchmark confirms that the proposed method can be successfully extended to complex, multi-dimensional elastic-plastic problems.

The physics-informed neural network (PINN) faces significant challenges when approximating solutions to conservation laws, particularly in ensuring conservation and accurately resolving discontinuities. To address these limitations, we propose solution character-informed neural network (SCINN), a novel framework that incorporates the boundedness constraint, implicit solution form, and Rankine–Hugoniot condition into the loss function, thereby enforcing conservation properties. Furthermore, we integrate a physics-based adaptive refinement (PAR) strategy to dynamically prioritize training near discontinuities, substantially improving the network’s ability to capture sharp gradients. Numerical experiments are conducted on benchmark problems, including the inviscid Burgers equation, the Lighthill-Whitham-Richards (LWR) traffic flow model and the Buckley-Leverett problem. The results show that compared to existing PINN-based methods, SCINN delivers outstanding performance in solving scalar conservation laws with wave interactions and non-convex non-concave flux functions. Compared to conventional PINN, SCINN yields a maximum reduction of 98.7% in mean squared error (MSE).

This paper proposes a systematic and explicit quantum circuit framework for solving advection-diffusion equations with boundary conditions, based on the Linear Combination of Hamiltonian Simulations (LCHS) method. By employing the Finite Volume Method (FVM) combined with various flux construction schemes, we elaborate the design of quantum circuits tailored explicitly for Robin boundary conditions (including Dirichlet and Neumann boundary conditions as special cases) and periodic boundary conditions. In contrast to prior works on quantum simulation of advection-diffusion equations, we present a detailed error analysis for the linear combination of unitaries (LCU) induced by the constructed quantum circuits. A comprehensive gate complexity analysis demonstrates the quantum advantages over classical computing in high-dimensional scenarios. We simulate the proposed circuits on a fault-tolerant emulator, and numerical results validate the effectiveness of the proposed framework across homogeneous, inhomogeneous, and high-dimensional cases. The proposed framework is compatible with numerous spatial discretization methods and numerical schemes, extends naturally to other linear PDEs, and establishes a practical foundation for solving large-scale PDE problems on future fault-tolerant quantum computers.

This paper presents a symplectic Hamiltonian direct discontinuous Galerkin (DDG) method for approximating wave propagation problems, including the linear and semilinear wave equations. Within an auxiliary-variable-free DG framework, we prove that the symmetry of the numerical flux bilinear form is equivalent to the existence of a discrete Hamiltonian structure. It follows that methods such as the symmetric interior penalty method and the symmetric DDG (SDDG) method admit a discrete Hamiltonian structure, whereas schemes including the Baumann–Oden, DDG, and BR2 methods do not possess this property. Exploiting this structure, we construct fully discrete symplectic schemes by combining the SDDG spatial discretization with symplectic time integrators. We further derive error estimates for the SDDG method applied to semilinear wave equations, showing the optimal convergence rate for the displacement and the suboptimal convergence rate for the velocity. Numerical experiments validate the theoretical convergence rates and demonstrate that the symplectic Hamiltonian DDG method achieves superior long-time energy conservation and accuracy.

The characteristic curve (characteristic) serves as a powerful tool for predicting the temporal evolution of solutions to conservation laws. This paper proposes a framework for classifying compressible flow fields from the perspective of the local characteristic distribution of fluid governing equations. First, the Symbolic Regression (SR) method is employed to derive a 2D vortex indicator from data. Inspired by the SR result, the eigenvalue configurations of the gradient matrix of the characteristic eigenvalues along orthogonal directions in mathematics are found to exhibit a one-to-one correspondence with the localized flow structures in physics. Based on this discovery and analysis, this paper pioneers a unified theoretical framework for compressible flow field classification based on local characteristic distribution. The framework is generalizable to 3D scenario and enables the simultaneous identification of vortices, shock waves, expansion waves, and departure structures within a unified theoretical framework, with only a slight increase in computational cost. Extensive 2D and 3D numerical results demonstrate that compared to other classical methods, the framework shows superior robustness against noise, and is capable of detecting dissipative structures, such as weak shocks and small-scale vortices. Classifying flow fields from the perspective of local characteristic distribution offers valuable insights in comprehending the generation, existence, and dissipation mechanisms of flow structures governed by fluid equations, such as shocks and vortices.

In this paper, we propose a path-conservative finite volume scheme to address the non-conservative nature of governing equations for hypo-elastic plastic solids. By combining the Hooke’s law-preserving path with the Riemann solver, a novel class of piecewise paths is proposed. Furthermore, we process the model changes during elastic-plastic transitions by incorporating deviatoric stress iteration, establishing transition formulas, and formulating the elastic-plastic interface fluxes. Our method has been applied to both one-dimensional and two-dimensional elastic-plastic solid models. Numerical tests have demonstrated its effectiveness.

Lung macrophage polarization imbalance is an important cause of aggravated pulmonary inflammation. The gut microbiota metabolites short-chain fatty acids (SCFAs) are an important regulator of macrophage polarization. A high-calorie diet has been shown to aggravate pneumonia and delay recovery, especially in children. However, the underlying mechanisms remain unclear. Our previous studies showed that a high-calorie diet can disrupt the gut microbiota structure and SCFA metabolism to aggravate LPS-induced lung inflammatory damage in juvenile rats. In this study, we investigated whether pneumonia aggravated owing to a high-calorie diet is associated with SCFA-driven macrophage phenotype changes in distal lung tissues and related mechanisms. Our data revealed that a high-calorie diet significantly aggravated pulmonary inflammatory injury in juvenile mice with LPS-induced pneumonia and also increased lung tissue M1-like (CD206-CD86+)/M2-like (CD206+CD86-) macrophage polarization imbalance. We found that a high-calorie diet decreased SCFA levels in mouse stool, serum, and lung tissues, which was most pronounced for acetate. Furthermore, we found that acetate reduction mediated by a high-calorie diet exacerbated M1-like (CD206⁻CD86⁺)/M2-like (CD206⁺CD86⁻) macrophage polarization imbalance in the lung tissue of pneumonia model mice and was associated with inhibiting histone deacetylase (HDAC), rather than G-protein-coupled receptor 43 (GPR43) signaling. More critically, we found that acetate supplementation had the most significant impact on HDAC9 and HDAC10 in the lung macrophages of pneumonia model mice fed a high-calorie diet. Furthermore, overexpression of Hdac9 and Hdac10 significantly attenuated the improvement effects of acetate on lung tissue M1-like (CD206-CD86+)/M2-like (CD206+CD86-) macrophage polarization in pneumonia model mice fed a high-calorie diet, and this mechanism was associated with the HIF-1α–glycolysis axis. Taken together, we demonstrated that a high-calorie diet could cause acetate levels to decrease in mice with LPS-induced pneumonia. This decrease in acetate was associated with a diminished inhibitory effect on HDAC9/10, potentially contributing to upregulation of HIF-1α expression and increased glycolysis. These changes may be linked to an imbalance in M1-like (CD206−CD86+)/M2-like (CD206+CD86−) macrophage polarization and aggravate lung tissue inflammatory injury. Our findings show that acetate supplementation may be a potential treatment strategy to prevent and treat pneumonia and other infectious diseases.

In this work, we develop a finite volume weighted essentially non-oscillatory (WENO) scheme based on an exponential approximation space for solving the five-equation transport model of compressible two-medium flows. Compared to the scheme based on the conventional polynomial approximation space, the performance of the present scheme is improved by locally optimizing the shape parameter in the exponential function such that the reconstructed solution can attain one enhanced order of accuracy without the need of increasing the stencil size. More importantly, through analyzing the relationship between the shape parameter and the linear weights of the WENO approximation, we present a suitable upper bound for the shape parameter which is able to avoid the appearance of negative linear weights. Finally, with the aim of maintaining computational robustness under large density and pressure ratios, we design a bound-preserving and positivity-preserving limiting algorithm targeted for the non-polynomial approximation space based finite volume WENO scheme. Numerical results demonstrate that the present scheme achieves high-order accuracy, high resolution and strong robustness in simulating compressible two-medium flows.

In this work, we develop a multidomain hybrid discontinuous Galerkin (DG) method and finite difference (FD) method for solving two-dimensional compressible Navier-Stokes equations on the hybrid meshes. The direct discontinuous Galerkin (DDG) method and central difference (CD) scheme are utilized to discretize the viscous fluxes respectively. This approach combines the flexibility for the complex geometries of the DG method on the unstructured meshes, and the computational efficiency of the FD method on Cartesian grids. At the artificial interfaces between the DG subdomain and FD subdomain, the square ghost cells are generated and the weighted essentially non-oscillatory (WENO) interpolation is employed to reconstruct the degrees of freedom of these ghost cells. To ensure the accuracy in smooth regions and the correct position of the shock wave, the troubled cell indicator is adopted to determine the nonconservative or conservative coupling modes. The construction process of the numerical fluxes at the artificial interfaces is described specifically and the corresponding WENO interpolation coefficients are given in detail. Numerous numerical results demonstrate that the multidomain hybrid method achieves high-order accuracy in smooth regions, robustness in shock simulations, flexibility in handling complex geometries, and significant computational cost savings compared to the traditional DG method on the hybrid meshes.

Vortex-resolved simulations often require fine grids to capture multi-scale vortices, which leads to extremely high computational costs. This work proposes a direct discontinuous Galerkin method (DDGM) integrated with adaptive mesh refinement (AMR) for effectively reducing the expensive computational overhead of vortex-resolved simulations. First, a symbolic regression inspired vortex indicator is developed and generalized to arbitrary incompatible grids, enabling the concentration of computational grid resources on vortex regions. The most attractive advantage of this vortex indicator lies in its theoretically analyzed capability to capture dissipative small-scale vortices even on coarse grids, which is naturally compatible with AMR. Second, by integrating the AMR strategy guided by the present vortex indicator, an efficient and vortex-resolved DDGM is established. Numerical results validate that the vortex-resolved DDGM with AMR is able to resolve vortices of different scales in complex flows with a slight increase in computational overhead, providing a promising numerical method for exploring vortex-induced lift mechanisms and underlying flow physics in aerospace applications.

Based on recent work by He et al. (Computer Physics Communications 286 (2023): 108660), this work develops a robust and efficient workflow for 3D aerodynamic shape optimization (ASO) based on the discontinuous Galerkin method (DGM) with solution remapping technique. It is theoretically found and numerically validated through 2D cases that the DGMs can provide more precise adjoint-enabled gradients even on a coarse mesh as compared with the FVMs under coequal computational costs. A number of 2D and 3D intricate cases are tested to illustrate the potential advantages of the DGM-based ASO workflow in terms of computational efficiency and final optimization results. The results demonstrate that the solution remapping technique can save around 50%$$ 50\% $$ of the computational time. Moreover, the DGM-based workflow extends the capability to explore the design space, leading to aerodynamic shapes with superior performance.

This paper implements the CUDA and hybrid CUDA/MPI parallel computation based on GPGPU heterogeneous parallel strategies for the direct discontinuous method (DDG) on 3D unstructured grids. The direct discontinuous Galerkin method inherits the compactness of the discontinuous Galerkin (DG) method, making it well-suited for large-scale parallelization. Firstly, we present the full single-GPU implementation of the three-dimensional (3D) DDG method with cell-level parallelism and face-level parallelism. Herein, all the numerical operators including volume integration, face integration (numerical fluxes), conservation variables calculation, and time iteration, are implemented by designing the corresponding kernel functions. Especially, we implement several key memory access optimization strategies, which are crucial for performance improvement. Operators merging and shared memory utilizing reduces the number of global access. Such memory Coalescing and data structure reconstruction apparently enhances the efficiency of global memory access. To align with data access pattern, we employ atomic operations to eliminate data race conditions. Furthermore, we propose a full hybrid GPU/CPU heterogeneous parallel strategy to implement multi-GPU parallelization of the DDG method, where asynchronization optimization is introduced to fully overlap communication and computation and basically eliminates the communication overhead. Finally, several numerical tests are conducted on Tesla V100 Cards to show performance of the parallelization. In addition, we utilize the NVIDIA performance testing tool, nvprof, to evaluate multiple metrics of the kernel functions and conduct a detailed analysis of the results. In the tests of parallel scalability, the weak scaling efficiency achieves 97% from 4 to 32 GPU cards, and the strong scaling efficiency is 90% from 1 to 8 GPU cards.

Physics-informed Neural Network (PINN) faces significant challenges when approximating solutions to conservation laws, particularly in ensuring conservation and accurately resolving discontinuities. To address these limitations, we propose Conservation Law-informed Neural Network (CLINN), a novel framework that incorporates the boundedness constraint, implicit solution form, and Rankine-Hugoniot condition of scalar conservation laws into the loss function, thereby enforcing exact conservation properties. Furthermore, we integrate a residual-based adaptive refinement (RAR) strategy to dynamically prioritize training near discontinuities, substantially improving the network's ability to capture sharp gradients. Numerical experiments are conducted on benchmark problems, including the inviscid Burgers equation, the Lighthill-Whitham-Richards (LWR) traffic flow model, and the Buckley-Leverett problem. Results demonstrate that CLINN achieves superior accuracy in resolving solution profiles and discontinuity locations while reducing numeral oscillations. Compared to conventional PINN, CLINN yields a maximum reduction of 99.2

It is well-known that convergence to steady state is hard when a high order discontinuous Galerkin (DG) method is applied for transonic and supersonic flows in polynomial space even with post- processing such as limiters and positivity preservation. The method discretizes the hyperbolic conservation laws in space in advance to obtain a system of first-order ordinary differential equations in time. As a result, steady-state solution of the DG method is equivalent to the equilibrium point of this system. In this paper, we analyze the stability of DG methods in the view point of dynamical systems for the scalar conservation law. We show that the steady-state solution of the 3rd-order DG method is not always stable in the presence of shock waves, and then we propose an artificial viscosity to stabilize the DG method and shows that the artificial viscosity has to be order one of the mesh size to improve stability. To maintain higher order accuracy, the proposed artificial viscosity is only applied in the vicinities of shock waves together with a shock-wave indicator. Numerical results are given to verify theoretical analysis. Several transonic/supersonic flow test cases are also present to demonstrate the effectiveness of the present artificial viscosity.

Accurate and rapid prediction of flowfields is crucial for aerodynamic design. This work proposes a discontinuous Galerkin method (DGM) whose performance can be enhanced with increasing data, for rapid simulation of transonic flow around airfoils under various flow conditions. A lightweight and easily updatable data-driven model is built to predict roughly correct flowfield, and the DGM is then utilized as the CFD solver to refine the detailed flow structures and provide the corrected data. During the construction of the data-driven model, a zonal proper orthogonal decomposition (POD) method is designed to reduce the dimensionality of flowfield while preserving more near-wall flow features, and a weighted distance-based radial basis function (RBF) is constructed to enhance the generalization capability of flowfield prediction. Numerical results demonstrate that the lightweight data-driven model can predict the flowfield around a wide range of airfoils at Mach numbers ranging from 0.7 to 0.95 and angles of attack from -5 degrees to 5 degrees by learning from sparse data, and maintains high accuracy of the location and essential features of flow structures (such as shock waves). In addition, the data-driven model enhanced DGM is able to improve the computational efficiency and simulation robustness as compared to normal DGMs in simulating transonic inviscid/viscous airfoil flowfields on arbitrary grids, and further enables rapid aerodynamic evaluation of numerous sample points during the surrogate-based aerodynamic optimization.

We found that no convergence to the correct solution can happen when a popular method is applied to discretize the derivative appearing in the objective function for optimization problems with singularly perturbed ODE constraints. The non-convergence mentioned above can occur even if the error bound of the numerical solution of the state equation has nothing to do with the small parameter. We disclose that the underlying reason for non-convergence to the correct solution is an inaccurate derivative calculation in the objective function for a model problem, which is solvable mathematically. To ensure correct convergence regardless of the small parameter, we propose an entirely exponential-type scheme for solving the optimization problem, in which an exponential-type scheme is used for the derivative in the objective function, together with an exponential-type finite difference scheme for the state equation. Both theoretical analysis and numerical experiments can verify the correct convergence of E2S in solving the singularly perturbed equationconstrained optimization problem.

In this paper, we discuss the direct discontinuous Galerkin (DDG) method combined with two specific high-order explicit-implicit-null (EIN) time discretizations for solving the compressible Navier-Stokes (CNS) equations. This paper presents the EIN method whose basic idea is to add and subtract an identical Laplacian operator on the right-hand side of the considered equations, and then apply the implicit-explicit (IMEX) time-marching method to the equivalent equations. More specifically, the added term is treated implicitly while the rest of the terms are treated explicitly. The EIN method is designed to eliminate the severe time step restriction associated with explicit methods, without requiring any nonlinear iterative solver. Based on the Fourier method, we analyze the stability of the EIN-DDG schemes for the one-dimensional CNS equations, and further validate numerically that the stability criteria can be extended to the two-dimensional case. The numerical results demonstrate that our schemes achieve both stability and optimal orders of accuracy under a relaxed time-step restriction, provided that an appropriate coefficient is used for the Laplacian operator. Furthermore, the computational efficiency of different time discretizations, such as the strong stability-preserving Runge-Kutta (SSP-RK) and EIN methods, is evaluated and compared, demonstrating the advantages of the proposed schemes.

The hypothalamic-pituitary-adrenal (HPA) axis plays an important regulatory role in inflammatory responses to systemic or local infection in the host. A high-calorie diet, which can aggravate pediatric pneumonia and delay recovery, is intimately associated with HPA axis disorder; however, its underlying mechanisms remain unknown. This study examined whether the mechanism by which a high-calorie diet aggravates pneumonia is related to HPA axis disorder. In this study, juvenile rats were fed a high-calorie diet and/or nebulized with lipopolysaccharide (LPS) for model construction. Our data shows that a high-calorie diet increases interleukin-1 beta(IL-1β), interleukin-6 (IL-6) and tumor necrosis factor-alpha (TNF-α) levels in lung tissues and aggravates LPS-induced inflammatory injury in the lungs of juvenile rats. Additionally, we found that a high-calorie diet decreases the expression level of serum adrenocorticotropic hormone (ACTH) and corticosterone (CORT) in juvenile rats with pneumonia, resulting in HPA axis disorder. Hypothalamus proteomics and Western blot results proved that a high-calorie diet upregulated the expression level of hypothalamus hypoxia-inducible factor-1 alpha (HIF-1α) in juvenile rats with pneumonia, and this mechanism is associated with reduced HIF-1α ubiquitination. We further observed that HPA axis disorder was significantly abated and inflammatory damage in rat lung tissues was significantly alleviated after in vivo HIF-1α pathway inhibition. This shows that pneumonia aggravation by a high-calorie diet is associated with interference in the HIF-1α-mediated HPA axis. A high-calorie diet boosts HIF-1α signaling in the hypothalamus and exacerbates LPS-induced pneumonia by disrupting the HPA axis. This sheds light on lung inflammation and strengthens the lung-brain connection.