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$hp$-version $C^1$-continuous Petrov-Galerkin method for nonlinear second-order initial value problems with application to wave equations

Lina Wang, Mingzhu Zhang, Hongjiong Tian, Lijun Yi

TL;DR

It is shown that thehp-version of the CPG method superconverges at the nodal points of the time partition with regard to the time steps and approximation degrees.

Abstract

We introduce and analyze an $hp$-version $C^1$-continuous Petrov-Galerkin (CPG) method for nonlinear initial value problems of second-order ordinary differential equations. We derive a-priori error estimates in the $L^2$-, $L^\infty$-, $H^1$- and $H^2$-norms that are completely explicit in the local time steps and local approximation degrees. Moreover, we show that the $hp$-version $C^1$-CPG method superconverges at the nodal points of the time partition with regard to the time steps and approximation degrees. As an application, we apply the $hp$-version $C^1$-CPG method to time discretization of nonlinear wave equations. Several numerical examples are presented to verify the theoretical results.

$hp$-version $C^1$-continuous Petrov-Galerkin method for nonlinear second-order initial value problems with application to wave equations

TL;DR

It is shown that thehp-version of the CPG method superconverges at the nodal points of the time partition with regard to the time steps and approximation degrees.

Abstract

We introduce and analyze an -version -continuous Petrov-Galerkin (CPG) method for nonlinear initial value problems of second-order ordinary differential equations. We derive a-priori error estimates in the -, -, - and -norms that are completely explicit in the local time steps and local approximation degrees. Moreover, we show that the -version -CPG method superconverges at the nodal points of the time partition with regard to the time steps and approximation degrees. As an application, we apply the -version -CPG method to time discretization of nonlinear wave equations. Several numerical examples are presented to verify the theoretical results.
Paper Structure (26 sections, 1 theorem, 188 equations, 14 figures, 5 tables)

This paper contains 26 sections, 1 theorem, 188 equations, 14 figures, 5 tables.

Table of Contents

  1. Introduction
  2. $hp$-version of the $C^1$-continuous Petrov-Galerkin method
  3. Galerkin time discretization
  4. Algebraic formulation
  5. Error analysis
  6. Generalized Jacobi polynomials
  7. An auxiliary projection and its approximation properties
  8. Abstract error bounds
  9. $H^1$- and $H^2$-error estimates
  10. $L^2$-error estimate
  11. Nodal superconvergence estimates
  12. $L^{\infty}$-error estimate
  13. Application to nonlinear wave equations
  14. Model problem
  15. Galerkin semi-discretization in space
  16. ...and 11 more sections

Key Result

Corollary 3.1

Let $J:=(a,b), ~h=b-a$, $r\ge 3$ and $u\in H^{s_{0}+1}(J)$ with $s_0\geq 1$. Then we have for any real $s$, $1\leq s \leq \min\{r ,s_0\}$. Here, $\Gamma(\cdot)$ is the usual gamma function.

Figures (14)

  • Figure 5.1: Example 1: $H^1$-errors of the $h$-version.
  • Figure 5.2: Example 1: $H^1$-errors of the $p$-version.
  • Figure 5.3: Example 1: maximum function and derivative approximation errors at nodes versus $H^1$-errors of the $h$-version.
  • Figure 5.4: Example 1: maximum function and derivative approximation errors at nodes versus $H^1$-errors of the $p$-version.
  • Figure 5.5: Example 1: $C^1$-CPG method versus $C^0$-CPG method, $H^1$-errors of the $h$-version.
  • ...and 9 more figures

Theorems & Definitions (20)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 3.1
  • Remark 3.1
  • Remark 3.2
  • Corollary 3.1
  • proof
  • proof
  • Remark 3.3
  • ...and 10 more