August 2017, 22(6): 2389-2416. doi: 10.3934/dcdsb.2017122

Numerical solutions of viscoelastic bending wave equations with two term time kernels by Runge-Kutta convolution quadrature

Department of Mathematics, Hunan Normal University, Changsha 410081, Hunan, China

E-mail address: daxu@hunnu.edu.cn

Received  July 2015 Revised  February 2017 Published  March 2017

Fund Project: The author is supported by NSFC grant 11671131, the Construct Program of the Key Discipline in Hunan Province, Performance Computing and Stochastic Information Processing (Ministry of Education of China)

In this paper, we study the numerical solutions of viscoelastic bending wave equations
$u_{t}(x,~t)-\int_{0}^{t}[\beta_{1}(t-s)\,u_{xx}(x,~s) - \beta_{2}(t-s)\,u_{xxxx}(x,~s)]ds = f(x,~t),$
for
$ 0<x<1,~ 0<t\leq T $
, with self-adjoint boundary and initial value conditions, in which the functions
$ \beta_{1}(t) $
and
$ \beta_{2}(t) $
are completely monotonic on
$ (0,~\infty) $
and locally integrable, but not constant. The equations are discretised in space by the finite difference method and in time by the Runge-Kutta convolution quadrature. The stability and convergence of the schemes are analyzed by the frequency domain and energy methods. Numerical experiments are provided to illustrate the accuracy and efficiency of the proposed schemes.
Citation: Da Xu. Numerical solutions of viscoelastic bending wave equations with two term time kernels by Runge-Kutta convolution quadrature. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2389-2416. doi: 10.3934/dcdsb.2017122
References:
[1]

H. BrunnerJ. -P. Kauthen and A. Ostermann, Runge-Kutta time discretizations of parabolic Volterra integro-differential equations, J. Integ. Equ. Appl., 7 (1995), 1-16.

[2]

L. Banjai and Ch. Lubich, An error analysis of Runge-Kutta convolution quadrature, BIT Numer. Math., 51 (2011), 483-496.

[3]

L. BanjaiCh. Lubich and J. M. Melenk, Runge-Kutta convolution quadrature for operators arising in wave propagation, Numer. Math., 119 (2011), 1-20.

[4]

R. W. Carr and K. B. Hannsgen, A nonhomogeneous integro-differential equation in Hilbert space, SIAM J. Math. Anal., 10 (1979), 961-984.

[5]

R. W. Carr and K. B. Hannsgen, Resolvent formulas for a Volterra equation in Hilbert space, SIAM J. Math. Anal., 13 (1982), 459-483.

[6]

M. P. CalvoE. Cuesta and C. Palencia, Runge-Kutta convolution quadrature methods for well-posed equations with memory, Numer. Math., 107 (2007), 589-614.

[7] H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd edition, Charendon Press, Oxford, 1959.
[8]

G. Fairweather, Spline collocation methods for a class of hyperbolic partial integro-differential equations, SIAM J. Numer. Anal., 31 (1994), 444-460.

[9]

M. Lopez-Fernandez and S. Sauter, Generalized convolution quadrature based on RungeKutta methods, Numer. Math., 133 (2016), 743-779.

[10]

K. B. Hannsgen, Indirect Abelian theorems and a linear Volterra equation, Trans. Amer. Math. Soc., 142 (1969), 539-555.

[11]

K. B. Hannsgen and R. L. Wheeler, Uniform L1 behavior in classes of integro-differential equations with completely monotonic kernels, SIAM J. Math. Anal., 15 (1984), 579-594.

[12]

E. Hairer, S. P. Nϕrsett and G. Wanner, Solving Ordinary Differential Equations. Ⅰ: Nonstiff Problems, 2nd edition, Springer Series in Computational Mathematics, 8, Springer, Berlin, 1993.

[13]

E. Hairer and G. Wanner, Solving ordinary differential equations. Ⅱ: Stiff and DifferentialAlgebraic Problems, 2nd edition, Springer Series in Computational Mathematics, 14, Springer, Berlin, 1996.

[14]

K. B. Hannsgen and R. L. Wheeler, Complete monotonicity and resolvent of Volterra integrodifferential equations, SIAM J. Math. Anal., 13 (1982), 962-969.

[15]

X. Hu and L. Zhang, A compact finite difference scheme for the fourth-order fractional diffusion-wave system, Computer Phys. Communications, 182 (2011), 1645-1650.

[16]

X. Hu and L. Zhang, On finite difference methods for fourth-order fractional diffusion-wave and sub-diffusion systems, Appl. Math. Comput., 218 (2012), 5019-5034.

[17]

B. JinR. LazarovY. Liu and Z. Zhou, The Galerkin finite element method for a multi-term time-fractional diffusion equation, J. Comp. Phys., 281 (2015), 825-843.

[18]

C. H. Kim and U. J. Choi, Spectral collocation methods for a partial integro-differential equation with a weakly singular kernel, J. Austral. Math. Soc. Ser. B., 39 (1988), 408-430.

[19]

Ch. LubichI. H. Sloan and V. Thomée, Non-smooth data error estimates for approximations of an evolution equation with a positive-type memory term, Math. Comput., 65 (1996), 1-17.

[20]

F. LiuM. M. MeerschaertR. J. McCoughP. Zhuang and Q. Liu, Numerical methods for solving the multi-term time-fractional wave-diffusion equation, Fractional calculus and Appl. Anal., 16 (2013), 9-25.

[21]

J. C. López-Marcos, A difference scheme for a nonlinear partial integro-differential equation, SIAM J. Numer. Anal., 27 (1990), 20-31.

[22]

W. McLean and V. Thomée, Numerical solution of an evolution equation with a positive-type memory term, J. Austral. Math. Soc. Ser. B., 35 (1993), 23-70.

[23]

W. McLean and V. Thomée, Time discretization of an evolution equation via Laplace transforms, IMA J. Numer. Anal., 24 (2004), 439-463.

[24]

W. McLean and V. Thomée, Maximum-norm error analysis of a numerical solution via Laplace transformation and quadrature of a fractional order evolution equation, IMA J. Numer. Anal., 30 (2010), 208-230.

[25]

K. Mustapha and W. McLean, Discontinuous Galerkin method for an evolution equation with a memory term of positive type, Math. Comp., 78 (2009), 1975-1995.

[26]

K. Mustapha and W. McLean, Super-convergence of a discontinuous Galerkin method for fractional diffusion and wave equations, SIAM J. Numer. Anal., 51 (2013), 491-515.

[27]

K. Mustapha and D. Schötzau, Well-posedness of hp-version discontinuous Galerkin methods for fractional diffusion wave equations, IMA J. Numer. Anal., 34 (2014), 1426-1446.

[28]

W. McLean and V. Thomée, Numerical solution via laplace transforms of a fractional order evolution equation, J. Integral Eq. Appl., 22 (2010), 57-94.

[29]

R. D. Noren, Uniform L1 behavior in a class of linear Volterra equations, Quart. Appl. Math., 47 (1989), 547-554.

[30]

R. D. Noren, Uniform L1 behavior in classes of integro-differential equations with convex kernels, J. Integral Equations Appl., 1 (1988), 385-396.

[31]

R. D. Noren, Uniform L1 behavior for the solution of a Volterra equation with a parameter, SIAM J. Math. Anal., 19 (1988), 270-286.

[32]

A. K. PaniG. Fairweather and R. I. Fernandes, Alternating direction implicit orthogonal spline collocation methods for an evolution equation with a positive-type memory term, SIAM J. Numer. Anal., 46 (2008), 344-364.

[33]

J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics, Vol. , 87, Birkhäuser Verlag, Basel; Boston; Berlin, 1993.

[34]

M. Renardy, W. J. Hrusa and J. A. Nohel, Mathematical Problems in Viscoelasticity, Longman Essex, U. K. , 1987.

[35]

J. M. Sanz-Serna, A numerical method for a partial integro-differential equation, SIAM J. Numer. Anal., 25 (1988), 319-327.

[36]

T. Tang, A finite difference scheme for partial integro-differential equations with a weakly singular kernel, Appl. Numer. Math., 11 (1993), 309-319.

[37]

J. Tang and D. Xu, The global behavior of finite difference-spatial spectral collocation methods for a partial integro-differential equation with a weakly singular kernel, Numer. Math. Theor. Meth. Appl., 6 (2013), 556-570.

[38] D. V. Widder, The Laplace Transform, Princeton University Press, Princeton, NJ., 1946.
[39]

D. Xu, Uniform L1 error bounds for the semidiscrete solution of a Volterra equation with completely monotonic convolution kernel, Computers Math. Appl., 43 (2002), 1303-1318.

[40]

D. Xu, Uniform l1 behaviour for time discretization of a volterra equation with completely monotonic kernel:Ⅰ. Stability, IMA J. Numer. Anal., 22 (2002), 133-151.

[41]

D. Xu, Uniform l1 behaviour in the second order difference type method of a linear Volterra equation with completely monotonic kernel Ⅰ: stability, IMA J. Numer. Anal., 31 (2011), 1154-1180.

[42]

D. Xu, Uniform l1 behaviour for time discretization of a Volterra equation with completely monotonic kernel Ⅱ: Convergence, SIAM J. Numer. Anal., 46 (2008), 231-259.

[43]

D. Xu, Stability of the difference type methods for linear Volterra equations in Hilbert spaces, Numer. Math., 109 (2008), 571-595.

[44]

D. Xu, Numerical solution of evolutionary integral equations with completely monotonic kernel by Runge-Kutta convolution quadrature, Numer. Meth. Partial Diff. Eq., 31 (2015), 105-142.

[45]

D. Xu, The time discretization in classes of integro-differential equations with completely monotonic kernels; Weighted asymptotic stability, Sci. China Math., 56 (2013), 395-424.

[46]

Y. Yi and G. Fairweather, Orthogonal spline collocation methods for some partial integrodifferential equations, SIAM J. Numer. Anal., 29 (1992), 755-768.

[47]

H. YeF. LiuI. TurnerV. Anh and K. Burrage, Series expansion solutions for the multiterm time and space fractional partial differential equations in two-and three-dimensions, Eur. Phys. J. Special Topics, 222 (2013), 1901-1914.

show all references

References:
[1]

H. BrunnerJ. -P. Kauthen and A. Ostermann, Runge-Kutta time discretizations of parabolic Volterra integro-differential equations, J. Integ. Equ. Appl., 7 (1995), 1-16.

[2]

L. Banjai and Ch. Lubich, An error analysis of Runge-Kutta convolution quadrature, BIT Numer. Math., 51 (2011), 483-496.

[3]

L. BanjaiCh. Lubich and J. M. Melenk, Runge-Kutta convolution quadrature for operators arising in wave propagation, Numer. Math., 119 (2011), 1-20.

[4]

R. W. Carr and K. B. Hannsgen, A nonhomogeneous integro-differential equation in Hilbert space, SIAM J. Math. Anal., 10 (1979), 961-984.

[5]

R. W. Carr and K. B. Hannsgen, Resolvent formulas for a Volterra equation in Hilbert space, SIAM J. Math. Anal., 13 (1982), 459-483.

[6]

M. P. CalvoE. Cuesta and C. Palencia, Runge-Kutta convolution quadrature methods for well-posed equations with memory, Numer. Math., 107 (2007), 589-614.

[7] H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd edition, Charendon Press, Oxford, 1959.
[8]

G. Fairweather, Spline collocation methods for a class of hyperbolic partial integro-differential equations, SIAM J. Numer. Anal., 31 (1994), 444-460.

[9]

M. Lopez-Fernandez and S. Sauter, Generalized convolution quadrature based on RungeKutta methods, Numer. Math., 133 (2016), 743-779.

[10]

K. B. Hannsgen, Indirect Abelian theorems and a linear Volterra equation, Trans. Amer. Math. Soc., 142 (1969), 539-555.

[11]

K. B. Hannsgen and R. L. Wheeler, Uniform L1 behavior in classes of integro-differential equations with completely monotonic kernels, SIAM J. Math. Anal., 15 (1984), 579-594.

[12]

E. Hairer, S. P. Nϕrsett and G. Wanner, Solving Ordinary Differential Equations. Ⅰ: Nonstiff Problems, 2nd edition, Springer Series in Computational Mathematics, 8, Springer, Berlin, 1993.

[13]

E. Hairer and G. Wanner, Solving ordinary differential equations. Ⅱ: Stiff and DifferentialAlgebraic Problems, 2nd edition, Springer Series in Computational Mathematics, 14, Springer, Berlin, 1996.

[14]

K. B. Hannsgen and R. L. Wheeler, Complete monotonicity and resolvent of Volterra integrodifferential equations, SIAM J. Math. Anal., 13 (1982), 962-969.

[15]

X. Hu and L. Zhang, A compact finite difference scheme for the fourth-order fractional diffusion-wave system, Computer Phys. Communications, 182 (2011), 1645-1650.

[16]

X. Hu and L. Zhang, On finite difference methods for fourth-order fractional diffusion-wave and sub-diffusion systems, Appl. Math. Comput., 218 (2012), 5019-5034.

[17]

B. JinR. LazarovY. Liu and Z. Zhou, The Galerkin finite element method for a multi-term time-fractional diffusion equation, J. Comp. Phys., 281 (2015), 825-843.

[18]

C. H. Kim and U. J. Choi, Spectral collocation methods for a partial integro-differential equation with a weakly singular kernel, J. Austral. Math. Soc. Ser. B., 39 (1988), 408-430.

[19]

Ch. LubichI. H. Sloan and V. Thomée, Non-smooth data error estimates for approximations of an evolution equation with a positive-type memory term, Math. Comput., 65 (1996), 1-17.

[20]

F. LiuM. M. MeerschaertR. J. McCoughP. Zhuang and Q. Liu, Numerical methods for solving the multi-term time-fractional wave-diffusion equation, Fractional calculus and Appl. Anal., 16 (2013), 9-25.

[21]

J. C. López-Marcos, A difference scheme for a nonlinear partial integro-differential equation, SIAM J. Numer. Anal., 27 (1990), 20-31.

[22]

W. McLean and V. Thomée, Numerical solution of an evolution equation with a positive-type memory term, J. Austral. Math. Soc. Ser. B., 35 (1993), 23-70.

[23]

W. McLean and V. Thomée, Time discretization of an evolution equation via Laplace transforms, IMA J. Numer. Anal., 24 (2004), 439-463.

[24]

W. McLean and V. Thomée, Maximum-norm error analysis of a numerical solution via Laplace transformation and quadrature of a fractional order evolution equation, IMA J. Numer. Anal., 30 (2010), 208-230.

[25]

K. Mustapha and W. McLean, Discontinuous Galerkin method for an evolution equation with a memory term of positive type, Math. Comp., 78 (2009), 1975-1995.

[26]

K. Mustapha and W. McLean, Super-convergence of a discontinuous Galerkin method for fractional diffusion and wave equations, SIAM J. Numer. Anal., 51 (2013), 491-515.

[27]

K. Mustapha and D. Schötzau, Well-posedness of hp-version discontinuous Galerkin methods for fractional diffusion wave equations, IMA J. Numer. Anal., 34 (2014), 1426-1446.

[28]

W. McLean and V. Thomée, Numerical solution via laplace transforms of a fractional order evolution equation, J. Integral Eq. Appl., 22 (2010), 57-94.

[29]

R. D. Noren, Uniform L1 behavior in a class of linear Volterra equations, Quart. Appl. Math., 47 (1989), 547-554.

[30]

R. D. Noren, Uniform L1 behavior in classes of integro-differential equations with convex kernels, J. Integral Equations Appl., 1 (1988), 385-396.

[31]

R. D. Noren, Uniform L1 behavior for the solution of a Volterra equation with a parameter, SIAM J. Math. Anal., 19 (1988), 270-286.

[32]

A. K. PaniG. Fairweather and R. I. Fernandes, Alternating direction implicit orthogonal spline collocation methods for an evolution equation with a positive-type memory term, SIAM J. Numer. Anal., 46 (2008), 344-364.

[33]

J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics, Vol. , 87, Birkhäuser Verlag, Basel; Boston; Berlin, 1993.

[34]

M. Renardy, W. J. Hrusa and J. A. Nohel, Mathematical Problems in Viscoelasticity, Longman Essex, U. K. , 1987.

[35]

J. M. Sanz-Serna, A numerical method for a partial integro-differential equation, SIAM J. Numer. Anal., 25 (1988), 319-327.

[36]

T. Tang, A finite difference scheme for partial integro-differential equations with a weakly singular kernel, Appl. Numer. Math., 11 (1993), 309-319.

[37]

J. Tang and D. Xu, The global behavior of finite difference-spatial spectral collocation methods for a partial integro-differential equation with a weakly singular kernel, Numer. Math. Theor. Meth. Appl., 6 (2013), 556-570.

[38] D. V. Widder, The Laplace Transform, Princeton University Press, Princeton, NJ., 1946.
[39]

D. Xu, Uniform L1 error bounds for the semidiscrete solution of a Volterra equation with completely monotonic convolution kernel, Computers Math. Appl., 43 (2002), 1303-1318.

[40]

D. Xu, Uniform l1 behaviour for time discretization of a volterra equation with completely monotonic kernel:Ⅰ. Stability, IMA J. Numer. Anal., 22 (2002), 133-151.

[41]

D. Xu, Uniform l1 behaviour in the second order difference type method of a linear Volterra equation with completely monotonic kernel Ⅰ: stability, IMA J. Numer. Anal., 31 (2011), 1154-1180.

[42]

D. Xu, Uniform l1 behaviour for time discretization of a Volterra equation with completely monotonic kernel Ⅱ: Convergence, SIAM J. Numer. Anal., 46 (2008), 231-259.

[43]

D. Xu, Stability of the difference type methods for linear Volterra equations in Hilbert spaces, Numer. Math., 109 (2008), 571-595.

[44]

D. Xu, Numerical solution of evolutionary integral equations with completely monotonic kernel by Runge-Kutta convolution quadrature, Numer. Meth. Partial Diff. Eq., 31 (2015), 105-142.

[45]

D. Xu, The time discretization in classes of integro-differential equations with completely monotonic kernels; Weighted asymptotic stability, Sci. China Math., 56 (2013), 395-424.

[46]

Y. Yi and G. Fairweather, Orthogonal spline collocation methods for some partial integrodifferential equations, SIAM J. Numer. Anal., 29 (1992), 755-768.

[47]

H. YeF. LiuI. TurnerV. Anh and K. Burrage, Series expansion solutions for the multiterm time and space fractional partial differential equations in two-and three-dimensions, Eur. Phys. J. Special Topics, 222 (2013), 1901-1914.

Table 8.1.  The $ L_{2} $ errors and convergence rate of the 2-stage Radau IIA convolution quadrature of (8.1.1) with $ M=802 $, and $ t_{K}=1 $
K $ e_{K} $ Rate
2 $ 2.9416e-006 $ -
4 $ 2.6658e-007 $ $ 3.4640 $
8 $ 2.8401e-008 $ $ 3.2306 $
2 $ 2.9416e-006 $ -
8 $ 2.8401e-008 $ $ 3.3473 $
Theory $ 3.0000 $
K $ e_{K} $ Rate
2 $ 2.9416e-006 $ -
4 $ 2.6658e-007 $ $ 3.4640 $
8 $ 2.8401e-008 $ $ 3.2306 $
2 $ 2.9416e-006 $ -
8 $ 2.8401e-008 $ $ 3.3473 $
Theory $ 3.0000 $
Table 8.2.  The $ L_{2} $ errors and convergence rate of the 3-stage Radau IIA convolution quadrature of (8.1.1) with $ M=802 $ and $ t_{K}=1 $
K $ e_{K} $ Rate
2 $ 3.1196e-006 $ -
4 $ 9.0652e-008 $ $ 5.1049 $
8 $ 2.8981e-009 $ $ 4.9672 $
2 $ 3.1196e-006 $ -
8 $ 2.8981e-009 $ $ 5.0360 $
K $ e_{K} $ Rate
2 $ 3.1196e-006 $ -
4 $ 9.0652e-008 $ $ 5.1049 $
8 $ 2.8981e-009 $ $ 4.9672 $
2 $ 3.1196e-006 $ -
8 $ 2.8981e-009 $ $ 5.0360 $
Table 8.3.  The $ L_{2} $ errors and convergence rates of the 2-stage Radau IIA convolution quadrature of (8.1.1) with $ M=802 $, $ t_{K}=1 $, and $ f(x~t) = \frac{t^{5.5}}{\Gamma(6.5)} \sin(\pi x) e^{\pi x} (\pi x)^{2}(\pi-\pi x)^{2} $
K $ e_{K} $ Rate
2 $ 7.4612e-005 $ -
4 $ 6.7979e-006 $ $ 3.4562 $
8 $ 7.3481e-007 $ $ 3.2096 $
2 $ 7.4612e-005 $ -
8 $ 7.3481e-007 $ $ 3.3329 $
Theory $ 3.0000 $
K $ e_{K} $ Rate
2 $ 7.4612e-005 $ -
4 $ 6.7979e-006 $ $ 3.4562 $
8 $ 7.3481e-007 $ $ 3.2096 $
2 $ 7.4612e-005 $ -
8 $ 7.3481e-007 $ $ 3.3329 $
Theory $ 3.0000 $
Table 8.4.  The $ L_{2} $ errors and convergence rates of the 3-stage Radau IIA convolution quadrature of (8.1.1) with $ M=802 $, $ t_{K}=1 $, and $ f(x~t) = \frac{t^{5.5}}{\Gamma(6.5)} \sin(\pi x) e^{\pi x} (\pi x)^{2}(\pi-\pi x)^{2} $
K $ e_{K} $ Rate
2 $ 8.2319e-005 $ -
4 $ 2.2979e-006 $ $ 5.1628 $
8 $ 7.3507e-008 $ $ 4.9663 $
2 $ 8.2319e-005 $ -
8 $ 7.3507e-008 $ $ 5.0646 $
K $ e_{K} $ Rate
2 $ 8.2319e-005 $ -
4 $ 2.2979e-006 $ $ 5.1628 $
8 $ 7.3507e-008 $ $ 4.9663 $
2 $ 8.2319e-005 $ -
8 $ 7.3507e-008 $ $ 5.0646 $
Table 8.5.  The $ L_{2} $ errors and convergence rate of the 2-stage Radau IIA convolution quadrature of (5.1) with $ M=202 $, and $ t_{K}=1 $
K $ e_{K} $ Rate
2 $ 0.2789 $ -
4 $ 0.0088 $ $ 4.9861 $
8 $ 9.1479e-004 $ $ 3.2660 $
16 $ 1.4166e-004 $ $ 2.6910 $
2 $ 0.2789 $ -
8 $ 9.1479e-004 $ $ 4.1260 $
2 $ 0.2789 $ -
16 $ 1.4166e-004 $ $ 3.6477 $
4 $ 0.0088 $ -
16 $ 1.4166e-004 $ $ 2.9785 $
K $ e_{K} $ Rate
2 $ 0.2789 $ -
4 $ 0.0088 $ $ 4.9861 $
8 $ 9.1479e-004 $ $ 3.2660 $
16 $ 1.4166e-004 $ $ 2.6910 $
2 $ 0.2789 $ -
8 $ 9.1479e-004 $ $ 4.1260 $
2 $ 0.2789 $ -
16 $ 1.4166e-004 $ $ 3.6477 $
4 $ 0.0088 $ -
16 $ 1.4166e-004 $ $ 2.9785 $
Table 8.6.  The $ L_{2} $ errors and convergence rate of the 3-stage Radau IIA convolution quadrature of (5.1) with $ M=202 $ and $ t_{K}=1 $
K $ e_{K} $ Rate
2 $ 0.0592 $ -
4 $ 4.9980e-004 $ $ 6.8881 $
8 $ 1.5134e-005 $ $ 5.0455 $
2 $ 0.0592 $ -
8 $ 1.5134e-005 $ $ 5.9668 $
K $ e_{K} $ Rate
2 $ 0.0592 $ -
4 $ 4.9980e-004 $ $ 6.8881 $
8 $ 1.5134e-005 $ $ 5.0455 $
2 $ 0.0592 $ -
8 $ 1.5134e-005 $ $ 5.9668 $
Table 8.7.  The $ L_{2} $ errors and convergence rates of the 2-stage Radau IIA convolution quadrature of (5.1) with $ M=202 $, $ t_{K}=1 $, and $ u_{0}(x)= \sin(\pi x) e^{\cos(\pi x)} $
K $ e_{K} $ Rate
4 $ 0.0593 $ -
8 $ 0.0076 $ $ 2.9640 $
16 $ 8.7669e-004 $ $ 3.1159 $
4 $ 0.0593 $ -
16 $ 8.7669e-004 $ $ 3.0399 $
K $ e_{K} $ Rate
4 $ 0.0593 $ -
8 $ 0.0076 $ $ 2.9640 $
16 $ 8.7669e-004 $ $ 3.1159 $
4 $ 0.0593 $ -
16 $ 8.7669e-004 $ $ 3.0399 $
Table 8.8.  The $ L_{2} $ errors and convergence rates of the 3-stage Radau IIA convolution quadrature of (5.1) with $ M=202 $, $ t_{K}=1 $, and $ u_{0}(x)= (\sin(\pi x))^{3} $
K $ e_{K} $ Rate
2 $ 0.0664 $ -
4 $ 0.0027 $ $ 4.6206 $
8 $ 7.4457e-005 $ $ 5.1804 $
2 $ 0.0664 $ -
8 $ 7.4457e-005 $ $ 4.9003 $
K $ e_{K} $ Rate
2 $ 0.0664 $ -
4 $ 0.0027 $ $ 4.6206 $
8 $ 7.4457e-005 $ $ 5.1804 $
2 $ 0.0664 $ -
8 $ 7.4457e-005 $ $ 4.9003 $
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