From Physics Avoidance to Symmetry: Discretization as Model Construction in Computer Simulation

This paper addresses the problem of numerical methods: when a continuous problem is transformed into a discrete procedure so that it can be computed, which physical or conceptual aspects are preserved, modified, or lost? Discretizing a continuous model typically does more than “compute the same thing approximately”: it replaces a continuous mathematical object with a different one, defined on a different space, governed by different invariants, and controlled by different notions of stability. The paper draws on what Mark Wilson calls physics avoidance: strategies by which applied mathematicians and modelers reduce calculational complexity by encoding physical and conceptual commitments into the architecture of the mathematical problem. I argue that physics avoidance is better understood as a family of conceptual strategies with distinctive epistemic risks than as “approximation,” and that discretization in particular is a model transformation rather than an approximation scheme. Convergence analysis, rigorous as it is, answers a different question from structural preservation: a convergent scheme recovers the exact solution only in the limit of refinement, and says nothing about whether the discrete solutions actually computed, at finite resolution and over long times, retain the structural features that make them physically meaningful. Finally, the paper proposes symmetry-based methods as a more careful way of reducing the complexity of scientific models: where physics avoidance simplifies a problem by hiding calculational decisions in its formalization, symmetries identify the invariant structures, conservation principles, and fundamental relations that a discretization should preserve.