Pin Lyu

Xue-lei Lin, Pin Lyu, Michael K. Ng et al.

In this paper, a second order finite difference scheme is investigated for time-dependent one-side space fractional diffusion equations with variable coefficients. The existing schemes for the equation with variable coefficients have temporal convergence rate no better than second order and spatial convergence rate no better than first order, theoretically. In the presented scheme, the Crank-Nicolson temporal discretization and a second-order weighted-and-shifted Grünwald-Letnikov spatial discretization are employed. Theoretically, the unconditional stability and the second-order convergence in time and space of the proposed scheme are established under some conditions on the diffusion coefficients. Moreover, a Toeplitz preconditioner is proposed for linear systems arising from the proposed scheme. The condition number of the preconditioned matrix is proven to be bounded by a constant independent of the discretization step-sizes so that the Krylov subspace solver for the preconditioned linear systems converges linearly. Numerical results are reported to show the convergence rate and the efficiency of the proposed scheme.

Pin Lyu, Yuxiang Liang, Zhibo Wang

In this paper, we study a fast and linearized finite difference method to solve the nonlinear time-fractional wave equation with multi fractional orders. We first propose a discretization to the multi-term Caputo derivative based on the recently established fast L2-1σ formula and a weighted approach. Then we apply the discretization to construct a fully fast linearized discrete scheme for the nonlinear problem under consideration. The nonlinear term, which just fulfills the Lipschitz condition, will be evaluated on the previous time level. Therefore only linear systems are needed to be solved for obtaining numerical solutions. The proposed scheme is shown to have second-order unconditional convergence with respect to the discrete H1-norm. Numerical examples are provided to justify the efficiency.

Huacan Wang, Ziyi Ni, Shuo Zhang et al.

The ultimate goal of code agents is to solve complex tasks autonomously. Although large language models (LLMs) have made substantial progress in code generation, real-world tasks typically demand full-fledged code repositories rather than simple scripts. Building such repositories from scratch remains a major challenge. Fortunately, GitHub hosts a vast, evolving collection of open-source repositories, which developers frequently reuse as modular components for complex tasks. Yet, existing frameworks like OpenHands and SWE-Agent still struggle to effectively leverage these valuable resources. Relying solely on README files provides insufficient guidance, and deeper exploration reveals two core obstacles: overwhelming information and tangled dependencies of repositories, both constrained by the limited context windows of current LLMs. To tackle these issues, we propose RepoMaster, an autonomous agent framework designed to explore and reuse GitHub repositories for solving complex tasks. For efficient understanding, RepoMaster constructs function-call graphs, module-dependency graphs, and hierarchical code trees to identify essential components, providing only identified core elements to the LLMs rather than the entire repository. During autonomous execution, it progressively explores related components using our exploration tools and prunes information to optimize context usage. Evaluated on the adjusted MLE-bench, RepoMaster achieves a 110% relative boost in valid submissions over the strongest baseline OpenHands. On our newly released GitTaskBench, RepoMaster lifts the task-pass rate from 40.7% to 62.9% while reducing token usage by 95%. Our code and demonstration materials are publicly available at https://github.com/QuantaAlpha/RepoMaster.

Ziyi Ni, Huacan Wang, Shuo Zhang et al.

Beyond scratch coding, exploiting large-scale code repositories (e.g., GitHub) for practical tasks is vital in real-world software development, yet current benchmarks rarely evaluate code agents in such authentic, workflow-driven scenarios. To bridge this gap, we introduce GitTaskBench, a benchmark designed to systematically assess this capability via 54 realistic tasks across 7 modalities and 7 domains. Each task pairs a relevant repository with an automated, human-curated evaluation harness specifying practical success criteria. Beyond measuring execution and task success, we also propose the alpha-value metric to quantify the economic benefit of agent performance, which integrates task success rates, token cost, and average developer salaries. Experiments across three state-of-the-art agent frameworks with multiple advanced LLMs show that leveraging code repositories for complex task solving remains challenging: even the best-performing system, OpenHands+Claude 3.7, solves only 48.15% of tasks (recent progress has pushed the frontier further, with RepoMaster+Claude 3.5 achieving a new record of 62.96%). Error analysis attributes over half of failures to seemingly mundane yet critical steps like environment setup and dependency resolution, highlighting the need for more robust workflow management and increased timeout preparedness. By releasing GitTaskBench, we aim to drive progress and attention toward repository-aware code reasoning, execution, and deployment -- moving agents closer to solving complex, end-to-end real-world tasks. The benchmark and code are open-sourced at https://github.com/QuantaAlpha/GitTaskBench.

Seakweng Vong, Pin Lyu

We study a second order scheme for spatial fractional differential equations with variable coefficients. Previous results mainly concentrate on equations with diffusion coefficients that are proportional to each other. In this paper, by further study on the generating function of the discretization matrix, second order convergence of the scheme is proved for diffusion coefficients satisfying a certain condition but are not necessary to be proportional. The theoretical results are justified by numerical tests.

Pin Lyu, Seakweng Vong

We consider difference schemes for nonlinear time fractional Klein-Gordon type equations in this paper. A linearized scheme is proposed to solve the problem. As a result, iterative method need not be employed. One of the main difficulties for the analysis is that certain weight averages of the approximated solutions are considered in the discretization and standard energy estimates cannot be applied directly. By introducing a new grid function, which further approximates the solution, and using ideas in some recent studies, we show that the method converges with second-order accuracy in time.