arXiv subjects

Tiegang Liu

Publications and source records attributed to Tiegang Liu.

9 recordsLinked to original sources

arXiv2026

Symplectic Hamiltonian Direct Discontinuous Galerkin Method for Wave Propagation

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.

math.NA↗

arXiv2026

On admissible solutions to the coupled Riemann problem with heat-flux discontinuity

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.

math.AP↗

arXiv2026

Quantum circuits for the advection-diffusion equation with boundary conditions based on LCHS

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.

math.NA↗

arXiv2025

CLINN: Conservation Law Informed Neural Network for Approximating Discontinuous Solutions

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% in mean squared error (MSE).

math.NA↗

arXiv2024

A machine learning enhanced discontinuous Galerkin method for simulating transonic airfoil flow-fields

Accurate and rapid prediction of flow-fields is crucial for aerodynamic design. This work proposes a discontinuous Galerkin method (DGM) whose performance enhances with increasing data, for rapid simulation of transonic flow around airfoils under various flow conditions. A lightweight and continuously updated data-driven model is built offline to predict the roughly correct flow-field, and the DGM is then utilized 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 flow-field while preserving more near-wall flow features, and a weighted-distance radial basis function (RBF) is constructed to enhance the generalization capability of flow-field prediction. Numerical results demonstrate that the lightweight data-driven model can predict the flow-field around a wide range of airfoils at Mach numbers ranging from 0.7 to 0.95 and angles of attack from -5 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 machine learning (ML) enhanced DGM is able to significantly improve the computational efficiency and simulation robustness as compared to normal DGMs in simulating transonic inviscid/viscous airfoil flow-fields on arbitrary grids, and further enables rapid aerodynamic evaluation of numerous sample points during the surrogate-based aerodynamic optimization.

physics.flu-dyn↗

arXiv2024

Investigation of discontinuous Galerkin methods in adjoint gradient-based aerodynamic shape optimization

This work develops a robust and efficient framework of the adjoint gradient-based aerodynamic shape optimization (ASO) using high-order discontinuous Galerkin methods (DGMs) as the CFD solver. The adjoint-enabled gradients based on different CFD solvers or solution representations are derived in detail, and the potential advantage of DG representations is discovered that the adjoint gradient computed by the DGMs contains a modification term which implies information of higher-order moments of the solution as compared with finite volume methods (FVMs). A number of numerical cases are tested for investigating the impact of different CFD solvers (including DGMs and FVMs) on the evaluation of the adjoint-enabled gradients. The numerical results demonstrate that the DGMs can provide more precise adjoint gradients even on a coarse mesh as compared with the FVMs under coequal computational costs, and extend the capability to explore the design space, further leading to acquiring the aerodynamic shapes with more superior aerodynamic performance.

math.NA↗

arXiv2022

A well-balanced scheme for Euler equations with singular sources

Numerical methods for the Euler equations with a singular source are discussed in this paper. The stationary discontinuity induced by the singular source and its coupling with the convection of fluid presents challenges to numerical methods. We show that the splitting scheme is not well-balanced and leads to incorrect results; in addition, some popular well-balanced schemes also give incorrect solutions in extreme cases due to the singularity of source. To fix such difficulties, we propose a solution-structure based approximate Riemann solver, in which the structure of Riemann solution is first predicted and then its corresponding approximate solver is given. The proposed solver can be applied to the calculation of numerical fluxes in a general finite volume method, which can lead to a new well-balanced scheme. Numerical tests show that the discontinuous Galerkin method based on the present approximate Riemann solver has the ability to capture each wave accurately.

math.NA↗

arXiv2022

Riemann problem of Euler equations with singular sources

This paper is concerned with the Riemann problem of one-dimensional Euler equations with a singular source. The exact solution of this Riemann problem contains a stationary discontinuity induced by the singular source, which is different from all the simple waves in the Riemann solution of classical Euler equations. We propose an eigenvalue-based monotonicity criterion to select the physical curve of this stationary discontinuity. By including this stationary discontinuity as an elementary wave, the structure of Riemann solution becomes diverse, e.g. the number of waves is not fixed and interactions between two waves become possible. Under the double CRP framework, we prove all possible structures of the Riemann solution.

math.AP↗

arXiv2022

Riemann problem for constant flow with single-point heating source

This work focuses on the Riemann problem of Euler equations with global constant initial conditions and a single-point heating source, which comes from the physical problem of heating one-dimensional inviscid compressible constant flow. In order to deal with the source of Dirac delta-function, we propose an analytical frame of double classic Riemann problems(CRPs) coupling, which treats the fluids on both sides of the heating point as two separate Riemann problems and then couples them. Under the double CRPs frame, the solution is self-similar, and only three types of solution are found. The theoretical analysis is also supported by the numerical simulation. Furthermore, the uniqueness of the Riemann solution is established with some restrictions on the Mach number of the initial condition.

math.AP↗