perform numerical ode integration

This study systematically compares MATLAB, Mathematica, and Maple in solving ordinary differential equations (ODEs), partial differential equations (PDEs), and systems of differential equations. A unified benchmark suite—grounded in analytically tractable reference solutions—is employed to empirically evaluate the tools across five dimensions: syntactic usability, numerical accuracy, computational efficiency, visualization capability, and specialized solver functionality. Crucially, the work introduces a novel, problem-driven software selection framework that classifies tasks by intrinsic characteristics—including equation type, stiffness, and boundary condition complexity. Results indicate that Mathematica excels in symbolic solution derivation and medium-scale ODE accuracy; MATLAB demonstrates superior performance in large-scale numerical simulation and engineering-oriented PDE applications; and Maple offers distinctive advantages in special-function handling and analytic derivation. This is the first systematic, multidimensional comparative study of these major mathematical software platforms, thereby bridging a critical gap in computational tool evaluation and providing actionable, evidence-based guidance for scientific and engineering practice.

This work addresses the challenges of integrating sparse linear algebra libraries into scientific computing applications—such as computational fluid dynamics (CFD), power grid simulation, and cardiac electrophysiology—including poor maintainability, high cross-platform adaptation costs, and tight coupling between application code and low-level implementations. We propose a modular integration framework built upon Ginkgo, which achieves loose coupling via a unified abstract interface, explicit decoupling of algorithms from hardware backends, and runtime backend selection. From a software engineering perspective, the framework significantly reduces integration complexity while enhancing portability, testability, and long-term maintainability. Experimental evaluation demonstrates that the framework sustains high performance across heterogeneous platforms (CPU/GPU), shortens the hardware adaptation cycle, and enables efficient, sustainable multi-domain simulation.

This paper addresses the complexity and poor scalability of manually deriving recurrence relations for Taylor coefficients of composite functions in high-order Taylor methods for solving ordinary differential equation (ODE) initial value problems. We propose the Sub-ODEs method, which transforms built-in functions (e.g., exp, sin) into inline differential relations, automatically generating coefficient recurrences and eliminating per-function manual derivation. We provide the first rigorous proof that Sub-ODE systems—regardless of nesting depth—guarantee mathematical completeness of the Taylor coefficient recurrence. This shifts the function library extension paradigm from “one recurrence per function” to “a few ODE rules per function.” Leveraging symbolic automatic differentiation and differential-algebraic modeling, we implement a custom solver in MATLAB. Experiments demonstrate superior accuracy and larger maximum allowable step sizes compared to mainstream MATLAB ODE solvers, achieving a new balance between generality and computational efficiency for high-order Taylor methods.

To address the insufficient time-integration capabilities of the SUNDIALS numerical library in high-performance scientific computing, this project systematically extends its time-stepping solvers. Methodologically, it introduces three novel classes of single-step methods—low-storage Runge–Kutta (LSRK), symplectic structure-preserving block RK, and general-purpose operator-splitting schemes—alongside a new multi-rate adaptive step-size controller and explicit RK-based adjoint sensitivity analysis (filling a longstanding gap). It further enhances nonlinear solvers with Anderson acceleration and improves error handling and logging infrastructure. These contributions significantly improve efficiency, stability, and accuracy for large-scale transient simulations, enabling long-duration, high-fidelity, and multiphysics-coupled modeling. Validation across multiple HPC applications demonstrates speedups of 1.5–3× and markedly improved numerical robustness.

Scientific software selection frequently suffers from non-reproducible benchmarks due to multi-library, multi-metric evaluation and dynamic evolution—such as the introduction of new algorithms or modifications to test cases and evaluation criteria. This paper addresses numerical integration over arbitrary 2D/3D domains with implicit or parameterized boundaries (cut-cell quadrature), proposing the first automated benchmarking framework that systematically integrates CI/CD engineering practices into scientific computing workflows. The framework unifies GitHub Actions, Docker, Python-based scheduling, Jupyter-based report generation, and semantically versioned result archiving. It supports automated configuration, execution, visualization, and historical result comparison. It achieves >90% automation for benchmark tasks and regression detection; reduces integration time for new libraries or algorithms by 70%; and enables precise attribution of performance deviations to specific code commits. The framework significantly enhances reliability, reproducibility, and evolutionary adaptability in scientific software evaluation.

Modern computational fluid dynamics (CFD) urgently requires seamless integration of simulation into design, optimization, and data-driven workflows, confronting challenges in the co-design of physical models, numerical methods, heterogeneous hardware, and automatic differentiation. This work systematically evaluates the suitability of the Julia programming language for CFD, leveraging its unified language ecosystem, multiple dispatch, and type specialization to deeply integrate high performance, differentiability, and software composability. Empirical validation through distributed CPU/multi-GPU parallelism, performance-portable frameworks, and open-source CFD projects demonstrates the feasibility of native Julia-based CFD at scale and its advantages in differentiable workflows. Nevertheless, the maturity of Julia’s industrial toolchain still lags behind that of conventional languages.

This work addresses the efficient algebraic representation and factorization of linear ordinary differential operators over compatible derivation modules. By implementing differential operators as first-class objects in Scratchpad II, the approach supports standard notation and provides a unified treatment of left and right module structures. For operators with coefficients in a field or polynomial ring, it integrates Ore localization, pseudo-division, and construction of right fraction fields to enable left and right division, computation of greatest common divisors, least common multiples, and extended Euclidean algorithms. Furthermore, by combining Riccati equations with Newton polygon analysis, the method effectively characterizes the singularities of factors. This framework facilitates constructive factorization and algebraic manipulation of operators with constant, elementary, rational, and even matrix-valued coefficients.

Existing formal verification methods for neural ordinary differential equations (neural ODEs) suffer from limited accuracy and scalability, supporting only single-shot reachability analysis. This work proposes the first end-to-end verification framework for neural ODEs, integrating continuous-time mixed-monotonicity theory, interval-based reachability analysis, a counterexample-guided iterative refinement loop for input sets, and a parallel scheduling mechanism. The framework further introduces three heuristic strategies for input set partitioning. It supports diverse neural ODE architectures and safety specifications, demonstrating significant improvements over state-of-the-art tools NNV 2.0 and CORA on benchmark problems. Experimental results show that the approach substantially enhances both the precision and efficiency of verifying safety set inclusion and classification robustness for neural ODEs.

This work addresses a key limitation of conventional LLM-based PDE solvers, which implicitly embed numerical strategies within generated code, making pre-execution validation and post-failure correction challenging. To overcome this, the authors propose AutoPDE, the first framework to explicitly model solution strategies as revisable, decoupled objects separate from implementation code. AutoPDE employs a three-stage pipeline—PDE type identification, numerical method selection, and adaptive parameter tuning—augmented by low-overhead trial solves and a reusable skill library to construct and refine strategies prior to code generation. Evaluated on the PDE Agent Bench, AutoPDE achieves a 54.5% pass rate, outperforming the strongest baseline by 14.2 percentage points, thereby substantially improving both the reliability and interpretability of AI-driven PDE solving.

This work investigates whether pretrained image editing models can serve as a universal interface for solving diverse physical equations. The approach encodes both inputs and solutions of physical problems as images, incorporates lightweight adapters to embed scalar parameters, and trains the model under a unified architecture using numerical or analytical solutions across multiple equation types—including elliptic, heat, and Navier-Stokes equations. For the first time, it systematically demonstrates that general-purpose generative models can effectively represent both static and dynamic physical mappings, even capturing shocks and unstable phenomena, thereby expanding their applicability in scientific computing. Experiments across more than ten problem classes yield promising results, yet also reveal limitations of image-based representations in handling wide numerical ranges, enforcing constraints, and simulating long-term chaotic dynamics, such as those in the Kuramoto–Sivashinsky equation.