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Convolution quadratures based on block generalized Adams methods

Ling Liu, Junjie Ma

TL;DR

The paper addresses efficient and high-order evaluation of convolution integrals $(K(\partial_t)g)(t)$ by discretizing the underlying IVP $y'(t)=\lambda y(t)+g(t)$ with a novel block generalized Adams (BGA) method on a uniform grid, forming a convolution quadrature (BGACQ) that retains the grid simplicity of LMCQ while enabling RK-like high-order accuracy for hyperbolic kernels. A rigorous convergence theory is developed, showing that, under analytic and growth assumptions on $K$, the CQ error scales as $\mathcal{O}(h^{k_1+k_2+2})$ for $\mu<0$ and as $\mathcal{O}(h^{\min\{k_1+k_2+3-\mu,\;k_1+k_2+2\}})$ for $\mu\ge 0$, with MBGACQ providing corrections to recover full order when $g$ has nonzero initial derivatives. The method employs a generating function approach with $\Delta(\zeta)$, FFT-based weight computation, and explicit stability considerations ($|R_m(i\omega)|<1$, etc.), and is validated through numerical experiments on hyperbolic kernels, fractional integrals, and oscillatory problems, showing competitive dissipation and high accuracy. The BGACQ framework thus offers a flexible, efficient tool for time-domain integral equations, Volterra problems, and related fractional dynamics, with future directions including larger blocks, fast weight computation refinements, and time-varying steps.

Abstract

This paper studies a family of convolution quadratures, a numerical technique for efficient evaluation of convolution integrals. We employ the block generalized Adams method to discretize the underlying initial value problem, departing from the well-established approaches that rely on linear multistep formulas or Runge-Kutta methods. The convergence order of the proposed convolution quadrature can be dynamically controlled without requiring grid point adjustments, enhancing exibility. Through strategic selection of the local interpolation polynomial and block size, the method achieves high-order convergence for calculation of convolution integrals with hyperbolic kernels. We provide a rigorous convergence analysis for the proposed convolution quadrature and numerically validate our theoretical findings for various convolution integrals.

Convolution quadratures based on block generalized Adams methods

TL;DR

The paper addresses efficient and high-order evaluation of convolution integrals by discretizing the underlying IVP with a novel block generalized Adams (BGA) method on a uniform grid, forming a convolution quadrature (BGACQ) that retains the grid simplicity of LMCQ while enabling RK-like high-order accuracy for hyperbolic kernels. A rigorous convergence theory is developed, showing that, under analytic and growth assumptions on , the CQ error scales as for and as for , with MBGACQ providing corrections to recover full order when has nonzero initial derivatives. The method employs a generating function approach with , FFT-based weight computation, and explicit stability considerations (, etc.), and is validated through numerical experiments on hyperbolic kernels, fractional integrals, and oscillatory problems, showing competitive dissipation and high accuracy. The BGACQ framework thus offers a flexible, efficient tool for time-domain integral equations, Volterra problems, and related fractional dynamics, with future directions including larger blocks, fast weight computation refinements, and time-varying steps.

Abstract

This paper studies a family of convolution quadratures, a numerical technique for efficient evaluation of convolution integrals. We employ the block generalized Adams method to discretize the underlying initial value problem, departing from the well-established approaches that rely on linear multistep formulas or Runge-Kutta methods. The convergence order of the proposed convolution quadrature can be dynamically controlled without requiring grid point adjustments, enhancing exibility. Through strategic selection of the local interpolation polynomial and block size, the method achieves high-order convergence for calculation of convolution integrals with hyperbolic kernels. We provide a rigorous convergence analysis for the proposed convolution quadrature and numerically validate our theoretical findings for various convolution integrals.
Paper Structure (5 sections, 4 theorems, 98 equations, 5 figures, 7 tables)

This paper contains 5 sections, 4 theorems, 98 equations, 5 figures, 7 tables.

Table of Contents

  1. Introduction
  2. BGA for IVP \ref{['IVP']}
  3. BGACQ for CI \ref{['CI']}
  4. Numerical Experiments
  5. Conclusions

Key Result

Lemma 2.1

\newlabelAdaErr Assume the function $g$ in CI CI possesses a continuous derivative of order at least $k_1+k_2+2$ over $[0,T],$ and satisfies the condition Then the error $q^n_j$ generated by the quadrature rule defined in Eq. Qerr satisfies

Figures (5)

  • Figure 2.1: Boundaries of the stability regions for BGAs with $m=m^*$ (left) and $m=12$ (right).
  • Figure 4.1: Comparison between pointwise errors of BGACQ$_{1,2}^5$ (left) and MBGACQ$_{1,2}^5$ (right) for CI \ref{['CI2ex']} with $\alpha=0.5$ and various $N.$
  • Figure 4.2: Comparison of numerical solutions computed by CQs for Eq. \ref{['IEex']} with 120 quadrature nodes.
  • Figure 4.3: Comparison of numerical solutions computed by BGACQ$_{1,1}^{4}$ and GRKCQ$_4$ for Eq. \ref{['IEex']} with 120 quadrature nodes.
  • Figure 4.4: Comparison of dissipative and conservative CQs for CI \ref{['CI5ex']}. The left part shows the relative errors as the quadrature nodes increase. The right part shows the relative errors as $\omega$ increases.

Theorems & Definitions (10)

  • Lemma 2.1
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • Example 1
  • Example 2
  • Example 3
  • Example 4