When I read statements like this, I wonder if this is related to optimal policy conditions required for infinitely lived Bellman equations to have global and per-period policies in alignment
Trying to timestep through a constraint that should hold instantaneously creates artificial numerical difficulties. The Bellman equation's value iteration is actually another example of this same pattern...
[1] Discounting: The discount factor β ∈ (0,1) is crucial. It ensures convergence of the value function and prevents “infinite accumulation” problems.
[2] Compactness of state/action sets: The feasible action correspondence Γ(x) is nonempty, compact-valued, and upper hemicontinuous in the state x. The state space X is compact (or at least the feasible set is bounded enough to avoid unbounded payoffs).
[3] Continuity: The return (or reward) function u(x,a) is continuous in (x,a). The transition law f(x,a) is continuous in (x,a).
[4] Bounded rewards: u(x,a) is bounded (often assumed continuous and bounded). This keeps the Bellman operator well-defined and ensures contraction mapping arguments go through.
[5] Contraction mapping property: With discounting and bounded payoffs, the Bellman operator is a contraction on the space of bounded continuous functions. This guarantees existence and uniqueness of the value function V.
[6] Measurable selection for policies: Under the above continuity and compactness assumptions, the maximum in the Bellman equation is attained, and there exists a measurable policy function g(x) that selects optimal actions.
https://en.wikipedia.org/wiki/Projection_method_(fluid_dynam...
A disadvantage is that you get a splitting error term that tends to dominate, so you gain nothing from using higher-order methods in time.
Of course, make sure you've done a thorough literature search, and that your paper is written from the perspective of "what is the contribution of this paper to the literature?", since most people reading your work will not read it in isolation: it'll be the hundredth or thousandth paper they've skimmed, trying to find those dozen papers relevant to their work.
That said, posting on arXiv can't be done unless someone vouches for you, which might be difficult or not for an independent person.
I think the best bet would be to submit your paper directly to a journal. However the paper in GP is unlikely to be published by any reputable journal. One direct feedback: if you can't explain at the start why your paper is relevant to current researcher's then why should anyone care? A sniff test for this is that you discuss in the introduction recent papers which have tried to solve the same or similar problems. But GP paper's references are over two decades old.
I publish to vixra... math is math.
Trying to handle instantaneous constraints and propagating modes with a single timestepper is often suboptimal.
I developed a framework that systematically separates these components: using direct elliptic solvers for constraints and explicit methods for flux evolution. The resulting algorithms are both more stable and more efficient than unified implicit approaches.
The key insight is that many systems (EM, GR, fluids, quantum mechanics) share the same pattern: - An elliptic constraint equation (solved directly, not timestepped) - A continuity law for charge/mass/probability flux - Wave-like degrees of freedom (handled with explicit methods)
Witgh this structure, you can avoid the stiffness issues entirely rather than trying to power through them with implicit methods.
Paper: https://zenodo.org/records/16968225