# American Institute of Mathematical Sciences

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

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 , 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. Brunner, J. -P. Kauthen and A. Ostermann, Runge-Kutta time discretizations of parabolic Volterra integro-differential equations, J. Integ. Equ. Appl., 7 (1995), 1-16. Google Scholar [2] L. Banjai and Ch. Lubich, An error analysis of Runge-Kutta convolution quadrature, BIT Numer. Math., 51 (2011), 483-496. Google Scholar [3] L. Banjai, Ch. Lubich and J. M. Melenk, Runge-Kutta convolution quadrature for operators arising in wave propagation, Numer. Math., 119 (2011), 1-20. Google Scholar [4] R. W. Carr and K. B. Hannsgen, A nonhomogeneous integro-differential equation in Hilbert space, SIAM J. Math. Anal., 10 (1979), 961-984. Google Scholar [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. Google Scholar [6] M. P. Calvo, E. Cuesta and C. Palencia, Runge-Kutta convolution quadrature methods for well-posed equations with memory, Numer. Math., 107 (2007), 589-614. Google Scholar [7] H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd edition, Charendon Press, Oxford, 1959. Google Scholar [8] G. Fairweather, Spline collocation methods for a class of hyperbolic partial integro-differential equations, SIAM J. Numer. Anal., 31 (1994), 444-460. Google Scholar [9] M. Lopez-Fernandez and S. Sauter, Generalized convolution quadrature based on RungeKutta methods, Numer. Math., 133 (2016), 743-779. Google Scholar [10] K. B. Hannsgen, Indirect Abelian theorems and a linear Volterra equation, Trans. Amer. Math. Soc., 142 (1969), 539-555. Google Scholar [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. Google Scholar [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.Google Scholar [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.Google Scholar [14] K. B. Hannsgen and R. L. Wheeler, Complete monotonicity and resolvent of Volterra integrodifferential equations, SIAM J. Math. Anal., 13 (1982), 962-969. Google Scholar [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. Google Scholar [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. Google Scholar [17] B. Jin, R. Lazarov, Y. Liu and Z. Zhou, The Galerkin finite element method for a multi-term time-fractional diffusion equation, J. Comp. Phys., 281 (2015), 825-843. Google Scholar [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. Google Scholar [19] Ch. Lubich, I. 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. Google Scholar [20] F. Liu, M. M. Meerschaert, R. J. McCough, P. 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. Google Scholar [21] J. C. López-Marcos, A difference scheme for a nonlinear partial integro-differential equation, SIAM J. Numer. Anal., 27 (1990), 20-31. Google Scholar [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. Google Scholar [23] W. McLean and V. Thomée, Time discretization of an evolution equation via Laplace transforms, IMA J. Numer. Anal., 24 (2004), 439-463. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [29] R. D. Noren, Uniform L1 behavior in a class of linear Volterra equations, Quart. Appl. Math., 47 (1989), 547-554. Google Scholar [30] R. D. Noren, Uniform L1 behavior in classes of integro-differential equations with convex kernels, J. Integral Equations Appl., 1 (1988), 385-396. Google Scholar [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. Google Scholar [32] A. K. Pani, G. 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. Google Scholar [33] J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics, Vol. , 87, Birkhäuser Verlag, Basel; Boston; Berlin, 1993.Google Scholar [34] M. Renardy, W. J. Hrusa and J. A. Nohel, Mathematical Problems in Viscoelasticity, Longman Essex, U. K. , 1987.Google Scholar [35] J. M. Sanz-Serna, A numerical method for a partial integro-differential equation, SIAM J. Numer. Anal., 25 (1988), 319-327. Google Scholar [36] T. Tang, A finite difference scheme for partial integro-differential equations with a weakly singular kernel, Appl. Numer. Math., 11 (1993), 309-319. Google Scholar [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. Google Scholar [38] D. V. Widder, The Laplace Transform, Princeton University Press, Princeton, NJ., 1946. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [43] D. Xu, Stability of the difference type methods for linear Volterra equations in Hilbert spaces, Numer. Math., 109 (2008), 571-595. Google Scholar [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. Google Scholar [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. Google Scholar [46] Y. Yi and G. Fairweather, Orthogonal spline collocation methods for some partial integrodifferential equations, SIAM J. Numer. Anal., 29 (1992), 755-768. Google Scholar [47] H. Ye, F. Liu, I. Turner, V. 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. Google Scholar show all references ##### References:  [1] H. Brunner, J. -P. Kauthen and A. Ostermann, Runge-Kutta time discretizations of parabolic Volterra integro-differential equations, J. Integ. Equ. Appl., 7 (1995), 1-16. Google Scholar [2] L. Banjai and Ch. Lubich, An error analysis of Runge-Kutta convolution quadrature, BIT Numer. Math., 51 (2011), 483-496. Google Scholar [3] L. Banjai, Ch. Lubich and J. M. Melenk, Runge-Kutta convolution quadrature for operators arising in wave propagation, Numer. Math., 119 (2011), 1-20. Google Scholar [4] R. W. Carr and K. B. Hannsgen, A nonhomogeneous integro-differential equation in Hilbert space, SIAM J. Math. Anal., 10 (1979), 961-984. Google Scholar [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. Google Scholar [6] M. P. Calvo, E. Cuesta and C. Palencia, Runge-Kutta convolution quadrature methods for well-posed equations with memory, Numer. Math., 107 (2007), 589-614. Google Scholar [7] H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd edition, Charendon Press, Oxford, 1959. Google Scholar [8] G. Fairweather, Spline collocation methods for a class of hyperbolic partial integro-differential equations, SIAM J. Numer. Anal., 31 (1994), 444-460. Google Scholar [9] M. Lopez-Fernandez and S. Sauter, Generalized convolution quadrature based on RungeKutta methods, Numer. Math., 133 (2016), 743-779. Google Scholar [10] K. B. Hannsgen, Indirect Abelian theorems and a linear Volterra equation, Trans. Amer. Math. Soc., 142 (1969), 539-555. Google Scholar [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. Google Scholar [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.Google Scholar [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.Google Scholar [14] K. B. Hannsgen and R. L. Wheeler, Complete monotonicity and resolvent of Volterra integrodifferential equations, SIAM J. Math. Anal., 13 (1982), 962-969. Google Scholar [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. Google Scholar [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. Google Scholar [17] B. Jin, R. Lazarov, Y. Liu and Z. Zhou, The Galerkin finite element method for a multi-term time-fractional diffusion equation, J. Comp. Phys., 281 (2015), 825-843. Google Scholar [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. Google Scholar [19] Ch. Lubich, I. 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. Google Scholar [20] F. Liu, M. M. Meerschaert, R. J. McCough, P. 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. Google Scholar [21] J. C. López-Marcos, A difference scheme for a nonlinear partial integro-differential equation, SIAM J. Numer. Anal., 27 (1990), 20-31. Google Scholar [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. Google Scholar [23] W. McLean and V. Thomée, Time discretization of an evolution equation via Laplace transforms, IMA J. Numer. Anal., 24 (2004), 439-463. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [29] R. D. Noren, Uniform L1 behavior in a class of linear Volterra equations, Quart. Appl. Math., 47 (1989), 547-554. Google Scholar [30] R. D. Noren, Uniform L1 behavior in classes of integro-differential equations with convex kernels, J. Integral Equations Appl., 1 (1988), 385-396. Google Scholar [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. Google Scholar [32] A. K. Pani, G. 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. Google Scholar [33] J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics, Vol. , 87, Birkhäuser Verlag, Basel; Boston; Berlin, 1993.Google Scholar [34] M. Renardy, W. J. Hrusa and J. A. Nohel, Mathematical Problems in Viscoelasticity, Longman Essex, U. K. , 1987.Google Scholar [35] J. M. Sanz-Serna, A numerical method for a partial integro-differential equation, SIAM J. Numer. Anal., 25 (1988), 319-327. Google Scholar [36] T. Tang, A finite difference scheme for partial integro-differential equations with a weakly singular kernel, Appl. Numer. Math., 11 (1993), 309-319. Google Scholar [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. Google Scholar [38] D. V. Widder, The Laplace Transform, Princeton University Press, Princeton, NJ., 1946. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [43] D. Xu, Stability of the difference type methods for linear Volterra equations in Hilbert spaces, Numer. Math., 109 (2008), 571-595. Google Scholar [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. Google Scholar [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. Google Scholar [46] Y. Yi and G. Fairweather, Orthogonal spline collocation methods for some partial integrodifferential equations, SIAM J. Numer. Anal., 29 (1992), 755-768. Google Scholar [47] H. Ye, F. Liu, I. Turner, V. 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. Google Scholar 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 $The$ 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 $The$ 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 $The$ 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 $The$ 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 $The$ 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 $The$ 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 $The$ 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 $ [1] Sihong Shao, Huazhong Tang. 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