Boundary treatment for high-order IMEX Runge-Kutta local discontinuous Galerkin schemes for multidimensional nonlinear parabolic PDEs

ArXiv ID: 2410.02927 “View on arXiv”

Authors: Unknown

Abstract

In this article, we propose novel boundary treatment algorithms to avoid order reduction when implicit-explicit Runge-Kutta time discretization is used for solving convection-diffusion-reaction problems with time-dependent Di-richlet boundary conditions. We consider Cartesian meshes and PDEs with stiff terms coming from the diffusive parts of the PDE. The algorithms treat boundary values at the implicit-explicit internal stages in the same way as the interior points. The boundary treatment strategy is designed to work with multidimensional problems with possible nonlinear advection and source terms. The proposed methods recover the designed order of convergence by numerical verification. For the spatial discretization, in this work, we consider Local Discontinuous Galerkin methods, although the developed boundary treatment algorithms can operate with other discretization schemes in space, such as Finite Differences, Finite Elements or Finite Volumes.

Keywords: Runge-Kutta, Implicit-Explicit Method, Convection-Diffusion-Reaction, Finite Volume Methods, Time Discretization, Quantitative Analytics

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats
• Why: The paper presents advanced mathematical concepts including IMEX Runge-Kutta schemes, Local Discontinuous Galerkin methods, and analysis of order reduction, demonstrating high math complexity. However, it focuses on theoretical numerical analysis and algorithmic improvements without mention of real financial data, backtesting, or implementation details relevant to quantitative finance.

  flowchart TD
    A["Research Goal: Develop boundary treatment algorithms for IMEX Runge-Kutta LDG schemes to prevent order reduction for nonlinear parabolic PDEs with time-dependent Dirichlet BCs"] --> B["Methodology: Construct novel boundary treatment algorithms<br>treat boundary values same as interior points at internal stages"]

    B --> C["Data/Inputs: Convection-Diffusion-Reaction Problems<br>Cartesian Meshes & Nonlinear Terms<br>Time-dependent Dirichlet BCs"]

    C --> D["Computational Process: Apply spatial discretization via Local Discontinuous Galerkin (LDG) methods<br>Integrate boundary treatment algorithms"]

    D --> E["Verification: Numerical verification of convergence orders"]

    E --> F["Key Findings/Outcomes: <br>1. Recover designed order of convergence<br>2. Effective for multidimensional problems<br>3. Compatible with other schemes<br>(FD, FEM, FV)"]