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Numerical solution of degenerate stochastic Kawarada equations via a semi-discretized approach

Joshua L Padgett, Qin Sheng

TL;DR

It is shown that key quenching solution features including the positivity and monotonicity are well preserved under modest restrictions and the numerical stability of the underlying splitting method is maintained without any additional restriction.

Abstract

The numerical solution of a highly nonlinear two-dimensional degenerate stochastic Kawarada equation is investigated. A semi-discretized approximation in space is comprised on arbitrary nonuniform grids. Exponential splitting strategies are then applied to advance solutions of the semi-discretized scheme over adaptive grids in time. It is shown that key quenching solution features including the positivity and monotonicity are well preserved under modest restrictions. The numerical stability of the underlying splitting method is also maintained without any additional restriction. Computational experiments are provided to not only illustrate our results, but also provide further insights into the global nonlinear convergence of the numerical solution.

Numerical solution of degenerate stochastic Kawarada equations via a semi-discretized approach

TL;DR

It is shown that key quenching solution features including the positivity and monotonicity are well preserved under modest restrictions and the numerical stability of the underlying splitting method is maintained without any additional restriction.

Abstract

The numerical solution of a highly nonlinear two-dimensional degenerate stochastic Kawarada equation is investigated. A semi-discretized approximation in space is comprised on arbitrary nonuniform grids. Exponential splitting strategies are then applied to advance solutions of the semi-discretized scheme over adaptive grids in time. It is shown that key quenching solution features including the positivity and monotonicity are well preserved under modest restrictions. The numerical stability of the underlying splitting method is also maintained without any additional restriction. Computational experiments are provided to not only illustrate our results, but also provide further insights into the global nonlinear convergence of the numerical solution.

Paper Structure

This paper contains 9 sections, 21 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Derivation of Scheme
  3. Positivity
  4. Monotonicity
  5. Stability
  6. Numerical Experiments
  7. Example 1
  8. Example 2
  9. Conclusions

Key Result

Lemma 2.1

Assume that $f(\varepsilon,u)\in C_c^2[0,1)$ and $\tau_k\rightarrow 0^+.$ Then for $\theta\in[0,1]$ we have where $p=2$ when $\theta = 1/2,$ and $p=1$ otherwise.

Theorems & Definitions (45)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 35 more