arXiv subjects

Jiachuan Cao

Publications and source records attributed to Jiachuan Cao.

6 recordsLinked to original sources

arXiv2026

Finite element exponential integration for rough solutions of nonlinear wave equations. Part I: Dirichlet boundary conditions on polygonal and polyhedral domains

We study a fully discrete scheme for nonlinear wave equations on general bounded polygonal/polyhedral domains with initial data $(u^0,v^0)\in H^γ(Ω)\times H^{γ-1}(Ω)$, $0<γ\le 1$, subject to the natural compatibility condition associated with the homogeneous Dirichlet boundary condition. The scheme combines an exponential Euler time integrator with a finite element spatial discretization. In contrast to existing low-regularity error analyses, which are mostly based on Fourier spectral discretizations, our approach applies to general bounded domains and finite element spatial discretizations. We prove rigorous error estimates for low-regularity solutions. The analysis is based on a frequency decomposition of the underlying elliptic operator, used solely as an analytical regularization device and not in the actual implementation, which allows low-regularity techniques to be extended beyond the Fourier spectral framework. The results also indicate that higher-order finite element methods remain advantageous in spatial approximation even for solutions of limited Sobolev regularity. Numerical experiments on different domains and with different polynomial degrees confirm the predicted convergence behavior.

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arXiv2026

Finite element exponential integration for rough solutions of nonlinear wave equations. Part II: Dynamic boundary conditions on curved domains

We study nonlinear wave equations with dynamic boundary conditions on smooth bounded domains and analyze a fully discrete approximation in the low-regularity regime. The method combines isoparametric bulk--surface finite elements of degree $k$ with an exponential integrator in time. Assuming only bounded energy of the exact solution, we prove convergence of the displacement--velocity pair in the weak norm $L^2(Ω;Γ)\times H^{-1}(Ω;Γ)$. The scheme achieves first-order convergence in time and spatial convergence of order $h^{2/3}$ for $k=1$ and $h^{(k+2)/(k+3)}$ for $k\ge 2$. In particular, these rates show that higher-order finite elements retain a provable asymptotic advantage even at low regularity. A central difficulty is that the continuous and discrete bulk--surface problems are posed on different geometries and must therefore be compared directly in weak norms. To address this, we develop a weak-norm framework for non-conforming geometries based on lift and adjoint-lift operators, combined with a frequency-decomposition argument. To the best of our knowledge, this is the first fully discrete low-regularity convergence result for nonlinear wave equations with dynamic boundary conditions in a non-conforming bulk--surface finite element setting. Numerical experiments confirm the predicted rates and illustrate the improved efficiency of higher-order methods.

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arXiv2026

A neural network method for scalar conservation laws with convergence rates for shock-wave solutions

We propose a new entropy-compatible neural network method for scalar hyperbolic conservation laws and establish, to our knowledge, the first explicit $L^1$ convergence rates in this setting that apply to piecewise smooth entropy solutions, including those with discontinuities. The method is based on a computable approximation of the Kružkov entropy residual that sits between the strong and weak forms of the entropy inequality. For piecewise smooth entropy solutions containing shocks, rarefactions, compound waves, regular shock interactions, and, in one space dimension, nondegenerate shock formation from smooth initial data, we construct explicit neural networks with provably small loss by combining shock-adapted continuous piecewise linear functions with known approximation properties of $\tanh$ neural networks. Together with entropy-based stability estimates, this gives rigorous $L^1$ error bounds for minimizers of the proposed loss. In particular, when the network size grows in proportion to the number of degrees of freedom of a space--time mesh of size $h$, the analysis recovers the classical Kuznetsov rate $O(h^{1/2})$ in shock-dominated cases. Numerical experiments in one and two space dimensions support the theory and suggest that the actual accuracy of the method can be better than the rate guaranteed by the analysis.

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arXiv2025

Computing rough solutions of the KdV equation below ${\bf L^2}$

We establish a novel numerical and analytical framework for solving the Korteweg--de Vries (KdV) equation in the negative Sobolev spaces, where classical numerical methods fail due to their reliance on high regularity and inability to control nonlinear interactions at low regularities. Numerical analysis is established by combining a continuous reformulation of the numerical scheme, the Bourgain-space estimates for the continuous reformulation, and a rescaling strategy that reduces the reformulated problem to a small initial value problem, which allow us to bridge a critical gap between numerical analysis and theoretical well-posedness by designing the first numerical method capable of solving the KdV equation in the negative Sobolev spaces. The numerical scheme is proved to have nearly optimal-order convergence with respect to the spatial degrees of freedom in the $H^{-\frac{1}{2}}$ norm for initial data in $H^s$, with $-\frac{1}{2} < s \leq 0$, a result unattainable by existing numerical methods.

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arXiv2024

Computing rough solutions of the stochastic nonlinear wave equation

The regularity of solutions to the stochastic nonlinear wave equation plays a critical role in the accuracy and efficiency of numerical algorithms. Rough or discontinuous initial conditions pose significant challenges, often leading to a loss of accuracy and reduced computational efficiency in existing methods. In this study, we address these challenges by developing a novel and efficient numerical algorithm specifically designed for computing rough solutions of the stochastic nonlinear wave equation, while significantly relaxing the regularity requirements on the initial data. By leveraging the intrinsic structure of the stochastic nonlinear wave equation and employing advanced tools from harmonic analysis, we construct a time discretization method that achieves robust convergence for initial values $(u^{0}, v^{0}) \in H^γ \times H^{γ-1}$ for all $γ> 0$. Notably, our method attains an improved error rate of $O(τ^{2γ-})$ in one and two dimensions for $γ\in (0, \frac{1}{2}]$, and $O(τ^{\max(γ, 2γ- \frac{1}{2}-)})$ in three dimensions for $γ\in (0, \frac{3}{4}]$, where $τ$ denotes the time step size. These convergence rates surpass those of existing numerical methods under the same regularity conditions, underscoring the advantage of our approach. To validate the performance of our method, we present extensive numerical experiments that demonstrate its superior accuracy and computational efficiency compared to state-of-the-art methods. These results highlight the potential of our approach to enable accurate and efficient simulations of stochastic wave phenomena even in the presence of challenging initial conditions.

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arXiv2024

Numerical approximation of discontinuous solutions of the semilinear wave equation

A high-frequency recovered fully discrete low-regularity integrator is constructed to approximate rough and possibly discontinuous solutions of the semilinear wave equation. The proposed method, with high-frequency recovery techniques, can capture the discontinuities of the solutions correctly without spurious oscillations and approximate rough and discontinuous solutions with a higher convergence rate than pre-existing methods. Rigorous analysis is presented for the convergence rates of the proposed method in approximating solutions such that $(u,\partial_{t}u)\in C([0,T];H^γ\times H^{γ-1})$ for $γ\in(0,1]$. For discontinuous solutions of bounded variation in one dimension (which allow jump discontinuities), the proposed method is proved to have almost first-order convergence under the step size condition $τ\sim N^{-1}$, where $τ$ and $N$ denote the time step size and the number of Fourier terms in the space discretization, respectively. Numerical examples are presented in both one and two dimensions to illustrate the advantages of the proposed method in improving the accuracy in approximating rough and discontinuous solutions of the semilinear wave equation. The numerical results are consistent with the theoretical results and show the efficiency of the proposed method.

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