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Summation-by-parts operators for general function spaces: The second derivative

Jan Glaubitz, Simon-Christian Klein, Jan Nordström, Philipp Öffner

Abstract

Many applications rely on solving time-dependent partial differential equations (PDEs) that include second derivatives. Summation-by-parts (SBP) operators are crucial for developing stable, high-order accurate numerical methodologies for such problems. Conventionally, SBP operators are tailored to the assumption that polynomials accurately approximate the solution, and SBP operators should thus be exact for them. However, this assumption falls short for a range of problems for which other approximation spaces are better suited. We recently addressed this issue and developed a theory for first-derivative SBP operators based on general function spaces, coined function-space SBP (FSBP) operators. In this paper, we extend the innovation of FSBP operators to accommodate second derivatives. The developed second-derivative FSBP operators maintain the desired mimetic properties of existing polynomial SBP operators while allowing for greater flexibility by being applicable to a broader range of function spaces. We establish the existence of these operators and detail a straightforward methodology for constructing them. By exploring various function spaces, including trigonometric, exponential, and radial basis functions, we illustrate the versatility of our approach. The work presented here opens up possibilities for using second-derivative SBP operators based on suitable function spaces, paving the way for a wide range of applications in the future.

Summation-by-parts operators for general function spaces: The second derivative

Abstract

Many applications rely on solving time-dependent partial differential equations (PDEs) that include second derivatives. Summation-by-parts (SBP) operators are crucial for developing stable, high-order accurate numerical methodologies for such problems. Conventionally, SBP operators are tailored to the assumption that polynomials accurately approximate the solution, and SBP operators should thus be exact for them. However, this assumption falls short for a range of problems for which other approximation spaces are better suited. We recently addressed this issue and developed a theory for first-derivative SBP operators based on general function spaces, coined function-space SBP (FSBP) operators. In this paper, we extend the innovation of FSBP operators to accommodate second derivatives. The developed second-derivative FSBP operators maintain the desired mimetic properties of existing polynomial SBP operators while allowing for greater flexibility by being applicable to a broader range of function spaces. We establish the existence of these operators and detail a straightforward methodology for constructing them. By exploring various function spaces, including trigonometric, exponential, and radial basis functions, we illustrate the versatility of our approach. The work presented here opens up possibilities for using second-derivative SBP operators based on suitable function spaces, paving the way for a wide range of applications in the future.
Paper Structure (23 sections, 1 theorem, 47 equations, 12 figures, 1 table)

This paper contains 23 sections, 1 theorem, 47 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Notation
  4. The first derivative
  5. The second derivative
  6. Existence and construction of second-derivative FSBP operators
  7. Existence of $\mathcal{F}$-exact second-derivative FSBP operators
  8. Existence of $(\mathcal{F} \bigoplus \mathcal{F}')$-exact first-derivative FSBP operators
  9. Construction of first- and second-derivative FSBP operators
  10. Examples
  11. Polynomial SBP operators
  12. Trigonometric FSBP operators
  13. Exponential SBP operators
  14. Radial basis function SBP operators
  15. Nullspaces of FSBP operators
  16. ...and 8 more sections

Key Result

Theorem 3.1

\newlabelthm:existence_second0 Let $\mathcal{F} \subset C^2([x_L,x_R])$ and $D_1 = P^{-1} Q$ be a first-derivative FSBP operator with $Q + Q^T = B$. The operator $D_2 = P^{-1}( B D_1 - D_1^T P D_1 )$ approximating $\partial_{xx}$ is an $\mathcal{F}$-exact second-derivative operator if and only if

Figures (12)

  • Figure 1: Preview of the possible advantage of using non-polynomial approximation spaces for specific problems. Illustrated here are the (numerical) solutions of a boundary layer problem and the viscous Burgers equation, which involve second derivatives. The solutions to both problems feature steep gradients that can be better approximated using exponential rather than traditional polynomial approximation spaces. We obtained the above numerical solutions using the FSBP-SAT scheme with polynomial ("poly") and exponential approximation spaces, $\mathcal{P}_2 =\mathrm{span}\{1,x,x^2\}$ and $\mathcal{E}_2 = \mathrm{span}\{1,x,e^x\}$, respectively. See \ref{['sub:bLayer', 'sub:Burgers']} for details.
  • Figure 1: (Numerical) solutions of the linear advection--diffusion equation \ref{['eq:periodc_advDif']} for $a=1$ and $\varepsilon = 10^{-5}$ with periodic initial and boundary data at time $t=1$ and mass and energy over time. The numerical solutions correspond to the FSBP-SAT method \ref{['eq:periodc_advDif_discr']} using the polynomial ("poly") and trigonometric ("trig") approximation space, $\mathcal{P}_{60}$ and $\mathcal{T}_{30}$. Both spaces have the same dimension of $K=61$ and use $N=61$ (Gauss--Lobatto) and $N=62$ (equidistant) grid points, respectively. The mass and energy profiles of the reference and numerical solutions using the trigonometric FSBP operators align.
  • Figure 2: (Numerical) solutions of the linear advection--diffusion equation \ref{['eq:periodc_advDif']} for $a=1$ and $\varepsilon = 10^{-2}$ with periodic initial and boundary data at time $t=0.1$ and mass and energy over time. We used a multi-block FSBP-SAT scheme with a polynomial and RBF approximation space, $\mathcal{P}_{2} = \mathrm{span}\{1,x,x^2\}$ and $\mathcal{R}_{3} = \mathrm{span}\{1,x,e^{x^2}\}$, respectively.
  • Figure 3: (Numerical) solutions of the linear advection--diffusion equation \ref{['eq:periodc_advDif']} for $a=1$ and $\varepsilon = 10^{-2}$ with periodic initial and boundary data at time $t=0.1$ and mass and energy over time. We used a multi-block FSBP-SAT scheme with $I=10$ blocks for the polynomial approximation space $\mathcal{P}_{2} = \mathrm{span}\{1,x,x^2\}$ and the RBF approximation space, $\mathcal{R}_{3} = \mathrm{span}\{1,x,e^{(x/\alpha)^2}\}$ for different shape parameters $\alpha=0.5,1,2,4,8,16$.
  • Figure 4: (Numerical) solutions of the two-dimensional linear advection--diffusion equation \ref{['eq:advDif_2D']} for $a_1=a_2=1$ and $\varepsilon_1 = \varepsilon_2 = 10^{-4}$ at time $t=1/4$. We used the tensor-product strategy combined with a multi-block FSBP-SAT scheme and $I=20$ blocks in each direction. We considered a polynomial and RBF approximation space, $\mathcal{P}_{2} = \mathrm{span}\{1,x,x^2\}$ and $\mathcal{R}_{3} = \mathrm{span}\{1,x,e^{20 x^2}\}$, respectively.
  • ...and 7 more figures

Theorems & Definitions (9)

  • Definition 2.1: First-derivative FSBP operators
  • Definition 2.2: Second-derivative FSBP operators
  • Remark 2.3
  • Remark 2.4
  • Theorem 3.1
  • Definition 4.1: The nullspace
  • Remark 5.1
  • Remark 5.2
  • Remark 5.3