Open research questions in Numerical methods for differential equations

What the literature leaves open

• The method requires additional computation. The memory overhead remained low, but may be a concern for very large problems. The study only considered a limited number of benchmark problems.

  Comparative analysis of the RK4 and hybrid RK4-feedforward neural network method for solving second-order ordinary differential equation initial value problems · 2026 · DOI

• Investigating the application of the hybrid RK4-FNN method to other types of differential equations. Exploring the use of other machine learning techniques to improve the accuracy and efficiency of numerical methods. Developing adaptive methods for selecting the optimal λ values.

  Comparative analysis of the RK4 and hybrid RK4-feedforward neural network method for solving second-order ordinary differential equation initial value problems · 2026 · DOI

• The curse of dimensionality in transport equations, which leads to significant computational costs. The need to handle multiple time and spatial scales in transport equations, which poses a challenge in terms of numerical resolution. The requirement for Hermiticity of the Hamiltonian in Hamiltonian simulation, which poses a challenge in solving ODEs and PDEs using Hamiltonian simulation.

  Quantum Simulation of Multiscale Linear Transport Equations via Schrödingerization and Exponential Integrators · 2026 · DOI

• The lack of quantum Hamiltonian simulation algorithms for multiscale linear transport equations that combine the Schrödingerization method with effective asymptotic-preserving schemes. The need for algorithms that can efficiently handle the curse of dimensionality and multiple time and spatial scales in transport equations.

  Quantum Simulation of Multiscale Linear Transport Equations via Schrödingerization and Exponential Integrators · 2026 · DOI

• The solution exhibits singularity at t = 0+ for its first time derivative, and at both t = 0+ and τ+ for its second time derivative. The complex interplay between memory and delay needs to be captured.

  Finite Element Method with Grünwald-Letnikov Type Approximation in Time for a Constant Time Delay Subdiffusion Equation · 2026 · DOI

• Traditional models are difficult to comprehensively capture the complex interplay between memory and delay. There is a need to develop a numerical scheme to solve the subdiffusion equation with constant time delay.

  Finite Element Method with Grünwald-Letnikov Type Approximation in Time for a Constant Time Delay Subdiffusion Equation · 2026 · DOI

• There is a need for efficient numerical methods for solving general fourth-order ordinary differential equations. The existing methods have limitations and are not efficient for solving certain types of problems.

  An Efficient Block Multistep Method for the Numerical Approximation of General Fourth-Order Ordinary Differential Equations · 2026 · DOI

• Balancing accuracy and stability in explicit integration methods. Reducing computational cost in nonlinear applications. Developing a method that remains explicit even when nonlinearities depend on first-derivative terms.

  A Novel Second-Order Explicit Integration Method for Nonlinear Ordinary Differential Equations in Dynamics · 2026 · DOI

• The method is limited to second-order ordinary differential equations. The stability of the method depends on the choice of parameters. The method may not be suitable for stiff systems where low-frequency modes dominate the response.

  A Novel Second-Order Explicit Integration Method for Nonlinear Ordinary Differential Equations in Dynamics · 2026 · DOI

• Existing methods have setbacks such as low rate of convergence and smaller stability regions. There is a need for a method that can solve second-order differential equations directly and efficiently.

  Adams-Type Block Hybrid Method for Direct Solutions of Second-Order Differential Equations · 2026 · DOI

• No comparison with other recent block hybrid methods beyond TSHBM (2017) and HBM (2023) is provided, limiting the scope of validation against contemporary approaches.

  Adams-Type Block Hybrid Method for Direct Solutions of Second-Order Differential Equations · 2026 · DOI

• The analysis assumes that the nonlinear algebraic system at each time step (u[0] = u0 - (r/ε²)B(α)f(u[0])) can be solved exactly via Newton iteration, but the effect of inexact nonlinear solver convergence and iteration error accumulation on the global L2-error bound remains unanalyzed.

  On the Conjecture of Stability Preservation in Arbitrary-Order Adams-Bashforth-Type Integrators · 2026 · DOI

• The choice of parameter α in the ABTI scheme does not affect the step size constraint according to the paper, but the impact of different α values on the error constant C in the L2-error bound C(τ^(q-1) + h^k) has not been characterized, leaving open how to optimize α for practical applications.

  On the Conjecture of Stability Preservation in Arbitrary-Order Adams-Bashforth-Type Integrators · 2026 · DOI

• The lack of structure-preserving variational integrators for non-autonomous Lagrangian systems. The need for high-order approximations of the true trajectory and the action integral. The challenge of capturing the long-term behavior and stability properties of non-autonomous systems.

  Structure-Preserving Time Integration of Non-Autonomous Lagrangian Systems Based on Prolongation–Collocation Variational Integrators · 2026 · DOI

• The nonlinearity of the PDE - The high dimensionality of the problem - The lack of efficient numerical methods

  Numerical Method for Nonlinear Kolmogorov PDEs via Sensitivity Analysis · 2026 · DOI

• The lack of efficient numerical methods for solving nonlinear Kolmogorov PDEs - The need for a sensitivity analysis based method

  Numerical Method for Nonlinear Kolmogorov PDEs via Sensitivity Analysis · 2026 · DOI

• Extension of the method to other types of boundary conditions. Application of the method to other fields, including biotechnology and pharmaceuticals.

  A high-accuracy symplectic scheme for a nonlinear transport problem · 2026 · DOI

• The advection-diffusion-reaction problem with non-homogeneous boundary conditions is challenging. There is a need for a high-accuracy symplectic scheme for solving the problem.

  A high-accuracy symplectic scheme for a nonlinear transport problem · 2026 · DOI

• The lack of numerical integrators that can preserve multiple invariants of conservative PDEs. The need for a novel approach to construct multiple invariants-preserving integrators.

  A Novel Semi-Analytical Multiple Invariants-Preserving Integrator for Conservative PDEs · 2026 · DOI

• The Krylov-Bogoliubov-Mitropolskii (KBM) method generally applies to weakly nonlinear and weakly damped systems, while the Harmonic Balance (HB) method often encounters convergence difficulties in time-dependent situations. There is a need for a hybrid analytical framework that can effectively model nonlinear oscillatory systems with strong damping and time-varying coefficients.

  APPROXIMATE DYNAMICAL ANALYSIS OF DAMPED NONLINEAR MECHANICAL SYSTEMS WITH TIME-VARYING PARAMETERS USING HYBRID ANALYTICAL METHODS · 2026 · DOI

• The computational cost associated with the numerical integration of large-scale matrix differential equations. The lack of robust and high-order accurate integrators for these equations.

  High-Order BUG Dynamical Low-Rank Integrators Based on Explicit Runge–Kutta Methods · 2026 · DOI

• The development of more efficient algorithms for the RK-BUG integrator. The application of the RK-BUG integrator to other types of differential equations.

  High-Order BUG Dynamical Low-Rank Integrators Based on Explicit Runge–Kutta Methods · 2026 · DOI

• The limited regularity of the non-linear part of the semilinear term. The development of a convergence analysis for the LBEFE and LCNFE methods.

  Linearly Implicit Finite Element Methods Approximating the Solution to the Nonlinear Schrödinger Equation with a Schamel-Type Nonlinearity · 2026 · DOI

• To extend the convergence analysis to other numerical methods for the nonlinear Schrödinger equation. To investigate the application of the methods to other physical and engineering problems.

  Linearly Implicit Finite Element Methods Approximating the Solution to the Nonlinear Schrödinger Equation with a Schamel-Type Nonlinearity · 2026 · DOI

• Beyond the second Bautin bifurcation, the extent of the bistable regime is delimited by the loci of limit points of cycles (LPC) emanating from the Bautin points.

  ff-bifbox: A scalable, open-source toolbox for bifurcation analysis of nonlinear PDEs · 2026 · DOI

Most-cited papers in Numerical methods for differential equations

• Equivalent Linearization Techniques · The Journal of the Acoustical Society of America · 1963 · 528 citations
• The automatic integration of ordinary differential equations · Communications of the ACM · 1971 · 483 citations
• Algorithm 407: DIFSUB for solution of ordinary differential equations [D2] · Communications of the ACM · 1971 · 267 citations
• Parallel methods for integrating ordinary differential equations · Communications of the ACM · 1964 · 160 citations
• Chebyshev polynomial solutions of linear differential equations · International Journal of Mathematical Education in Science and Technology · 1996 · 83 citations
• The effects of different modes of representation on the solution of one-step additive problems · Learning and Instruction · 2007 · 72 citations
• Numerical Integration of Ordinary Differential Equations · American Mathematical Monthly · 1926 · 56 citations
• Mixed Methods – Stand der Debatte und aktuelle Problemlagen · KZfSS Kölner Zeitschrift für Soziologie und Sozialpsychologie · 2017 · 54 citations
• Datenanalyse in der Mixed-Methods-Forschung · KZfSS Kölner Zeitschrift für Soziologie und Sozialpsychologie · 2017 · 45 citations
• AN EFFECTIVE COMPUTATIONAL APPROACH BASED ON HERMITE WAVELET GALERKIN FOR SOLVING PARABOLIC VOLTERRA PARTIAL INTEGRO DIFFERENTIAL EQUATIONS AND ITS CONVERGENCE ANALYSIS · Mathematical Modelling and Analysis · 2023 · 45 citations

Most recent work

• Convergence of an adaptive and conservative time-stepping scheme for the 2D nonlinear Schrödinger equation with wave operator · Journal of Computational and Applied Mathematics · 2026
• Finite Element Method with Grünwald-Letnikov Type Approximation in Time for a Constant Time Delay Subdiffusion Equation · Journal of Computational Mathematics · 2026
• An efficient high order solver for multidimensional nonlinear time-evolution (Burgers) equation based on Legendre pseudospectral method · Computational and Applied Mathematics · 2026
• High-Order BUG Dynamical Low-Rank Integrators Based on Explicit Runge–Kutta Methods · Journal of Scientific Computing · 2026
• New algorithms for Feynman integral reduction and epsilon-factorized differential equations · Physical Review D · 2026
• Average Energy Dissipation Rates of Implicit-Explicit Runge-Kutta Methods for Gradient Flow Problems · CSIAM Transactions on Applied Mathematics · 2026
• Regularized dynamical parametric approximation · Numerische Mathematik · 2026
• Optimal error estimates of two linearized finite difference schemes for the logarithmic Schrödinger equation · Applied Numerical Mathematics · 2026
• HOS-NWT: an open-source numerical wave tank based on the High-Order Spectral method · Computer Physics Communications · 2026
• A two-level direct solver for the hierarchical Poincaré-Steklov method · Computers & Mathematics with Applications · 2026

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