doi: 10.3934/cpaa.2021132
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Semi-discrete and fully discrete HDG methods for Burgers' equation

School of Mathematics, Sichuan University, Chengdu 610064, China

* Corresponding author

Received  January 2021 Revised  June 2021 Early access July 2021

Fund Project: This work was supported by National Natural Science Foundation of China (11771312)

This paper proposes semi-discrete and fully discrete hybridizable discontinuous Galerkin (HDG) methods for the Burgers' equation in two and three dimensions. In the spatial discretization, we use piecewise polynomials of degrees $ k \ (k \geq 1), k-1 $ and $ l \ (l = k-1; k) $ to approximate the scalar function, flux variable and the interface trace of scalar function, respectively. In the full discretization method, we apply a backward Euler scheme for the temporal discretization. Optimal a priori error estimates are derived. Numerical experiments are presented to support the theoretical results.

Citation: Zimo Zhu, Gang Chen, Xiaoping Xie. Semi-discrete and fully discrete HDG methods for Burgers' equation. Communications on Pure and Applied Analysis, doi: 10.3934/cpaa.2021132
References:
[1]

E. N. Aksan, A numerical solution of Burgers' equation by finite element method constructed on the method of discretization in time, Appl. Math. Comput., 170 (2005), 895-904.  doi: 10.1016/j.amc.2004.12.027.

[2]

R. Alexande, Diagonally Implicit Runge-Kutta Methods for Stiff O.D.E.'s, SIAM J. Numer. Anal., 14 (1977), 1006-1021.  doi: 10.1137/0714068.

[3]

A. AliG. Gardner and L. Gardner, A collocation solution for Burgers' equation using cubic B-spline finite elements, Comput. Method. Appl. M., 100 (1992), 325-337.  doi: 10.1016/0045-7825(92)90088-2.

[4]

P. Arminjon and C. Beauchamp, A finite element method for Burgers' equation in hydrodynamics, Int. J. num. Meth. Eng, 12 (1978), 415-428.  doi: 10.1002/nme.1620120304.

[5]

P. Arminjon and C. Beauchamp, Continuous and discontinuous finite element methods for Burgers' equation, Comput. Methods Appl. M., 25 (1981), 65-84.  doi: 10.1016/0045-7825(81)90069-4.

[6]

S. Brenner and L. Scott, The Mathematical Theory of Finite Element Methods, 3$^{rd}$ edition, Springer, New York, 2008. doi: 10.1007/978-0-387-75934-0.

[7]

J. Burgers, A mathematical model illustrating the theory of turbulence, Adv. Appl. Mech., 1 (1948), 171-199.  doi: 10.1016/S0065-2156(08)70100-5.

[8]

J. Burgers, Mathematical examples illustrating relations occuring in the theory of turbulent fluid motion, Springer, Dordrecht, 1995. doi: 10.1007/978-94-011-0195-0_10.

[9]

E. Burman, Error estimates for forward Euler shock capturing finite element approximations of the one-dimensional Burgers' equation, Math. Mod. Meth. Appl. S., 25 (2015), 2015-2042.  doi: 10.1142/S0218202515500517.

[10]

J. CaldwellP. Wanless and A. Cook, A finite element approach to Burgers' equation, Appl. Math. Model., 5 (1981), 189-193.  doi: 10.1016/0307-904X(81)90043-3.

[11]

G. Chen and X. Xie, A robust weak Galerkin finite element method for linear elasticity with strong symmetric stresses, Comput. Meth. Appl. Mat., 16 (2016), 389-408.  doi: 10.1515/cmam-2016-0012.

[12]

H. Chen and Z. Jiang, A characteristics-mixed finite element method for Burgers' equation, J. Appl. Math. Comput., 15 (2004), 29-51.  doi: 10.1007/BF02935745.

[13]

H. ChenP. Lu and X. Xu, A robust multilevel method for hybridizable discontinuous Galerkin method for the Helmholtz equation, J. Comput. Phys., 264 (2014), 133-151.  doi: 10.1016/j.jcp.2014.01.042.

[14]

Y. Chen and T. Zhang, A weak Galerkin finite element method for Burgers' equation, J. Comput. Appl. Math., 348 (2019), 103-109.  doi: 10.1016/j.cam.2018.08.044.

[15]

B. CockburnJ. Gopalakrishnan and N. Nguyen, Analysis of HDG methods for Stokes flow, Math. Comput., 80 (2011), 723-760.  doi: 10.1090/S0025-5718-2010-02410-X.

[16]

A. Dogan, A Galerkin finite element approach to Burgers' equation, Appl. Math. Comput., 157 (2004), 331-346.  doi: 10.1016/j.amc.2003.08.037.

[17]

R. Guzzi and L. Stefanutti, The Role of Airflow in Airborne Transmission of COVID 19, Int. J. Biol. Sci., 4 (2021), 121-131.  doi: 10.13133/2532-5876/17224.

[18]

Y. Han, H. Chen, X. Wang and X. Xie, EXtended HDG methods for second order elliptic interface problems, J. Sci. Comput., 84 (2020), 22. doi: 10.1007/s10915-020-01272-3.

[19]

J. Heywood and R. Rannacher, Finite-Element Approximation of the Nonstationary Navier-Stokes Problem. Part IV: Error Analysis for Second-Order Time Discretization, SIAM J. Numer. Anal., 27 (1990), 353-384.  doi: 10.1137/0727022.

[20]

X. HuP. Huang and X. Feng, Two-Grid Method for Burgers' Equation by a New Mixed Finite Element Scheme, Math. Model. Anal., 19 (2014), 1-17.  doi: 10.3846/13926292.2014.892902.

[21]

X. HuP. Huang and X. Feng, A new mixed finite element method based on the Crank-Nicolson scheme for Burgers' equation, Appl. Math-Czech., 61 (2016), 27-45.  doi: 10.1007/s10492-016-0120-3.

[22]

A. Hussein and H. Kashkool, Weak Galerkin finite element method for solving one-dimensional coupled Burgers' equations, J. Appl. Math. Comput., 63 (2020), 265-293.  doi: 10.1007/s12190-020-01317-8.

[23]

O. Karakashian and F. Pascal, Convergence of adaptive discontinuous Galerkin approximations of second order elliptic problems, SIAM J. Numer. Anal., 45 (2007), 641-665.  doi: 10.1137/05063979X.

[24]

S. KutluayA. Esen and I. Dag, Numerical solutions of the Burgers' equation by the least-squares quadratic B-spline finite element method, J. Comput. Appl. Math., 167 (2004), 21-33.  doi: 10.1016/j.cam.2003.09.043.

[25]

B. Li and X. Xie, Analysis of a family of HDG methods for second order elliptic problems, J. Comput. Appl. Math., 307 (2016), 37-51.  doi: 10.1016/j.cam.2016.04.027.

[26]

C. W. Lucchi, Improvement of MacCormack's scheme for Burgers' equation. Using a finite element method, Int. J. num. Meth. Eng, 15 (1980), 537-555.  doi: 10.1002/nme.1620150406.

[27]

R. Mittal and A. Tripathi, Numerical solutions of two-dimensional Burgers' equations using modified Bi-cubic B-spline finite elements, Eng. Comput., 32 (2015), 1275-1306.  doi: 10.1108/EC-04-2014-0067.

[28]

N. NguyenJ. Peraire and B. Cockburn, An implicit high-order hybridizable discontinuous Galerkin method for linear convection-diffusion equations, J. Comput. Phys., 228 (2009), 3232-3254.  doi: 10.1016/j.jcp.2009.01.030.

[29]

N. NguyenJ. Peraire and B. Cockburn, An implicit high-order hybridizable discontinuous Galerkin method for nonlinear convection-diffusion equations, J. Comput. Phys., 228 (2009), 8841-8855.  doi: 10.1016/j.jcp.2009.08.030.

[30]

T. ÖzişE. Aksan and A. Özdeş, A finite element approach for solution of Burgers' equation, Appl. Math. Comput., 139 (2003), 417-428.  doi: 10.1016/S0096-3003(02)00204-7.

[31]

A. PanyN. Nataraj and S. Singh, A new mixed finite element method for Burgers' equation, J. Appl. Math. Comput., 23 (2007), 43-55.  doi: 10.1007/BF02831957.

[32]

W. QiuJ. Shen and K. Shi, An HDG method for linear elasticity with strong symmetric stresses, Math. Comput., 87 (2016), 69-93.  doi: 10.1090/mcom/3249.

[33]

L. ShaoX. Feng and Y. He, The local discontinuous Galerkin finite element method for Burger's equation, Math. Comput. Model., 54 (2011), 2943-2954.  doi: 10.1016/j.mcm.2011.07.016.

[34]

D. ShiJ. Zhou and D. Shi, A new low order least squares nonconforming characteristics mixed finite element method for Burgers' equation, Appl. Math. Comput., 219 (2013), 11302-11310.  doi: 10.1016/j.amc.2013.05.037.

[35] Z. Shi and M. Wang, Finite Element Methods, Science Press, Beijing, 2013. 
[36]

H. SterckT. ManteuffelS. Mccormick and L. Olson, Numerical conservation properities of H(div)-conforming least-squares finite element methods for the Burgers equation, SIAM J. Sci. Comput., 26 (2005), 1573-1597.  doi: 10.1137/S1064827503430758.

[37]

R. Temam, Infinite-Dimensional Dynamical System in Mechanics and Physics, 2$^{nd}$ edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4684-0313-8.

[38]

Y. UçarN. Yaǧmurlu and İ. Çelikkaya, Operator splitting for numerical solution of the modified Burgers' equation using finite element method, Numer. Meth. Part. D. E., 35 (2019), 478-492.  doi: 10.1002/num.22309.

[39]

J. Warga, Optimal Control of Differential and Functional Equations, 1st edition, Academic Press, New York, 1972. doi: 10.1016/C2013-0-11669-8.

[40]

D. Winterscheidt and K. Surana, p-version least-squares finite element formulation of Burgers' equation, Int. J. num. Meth. Eng, 36 (2010), 3629-3646.  doi: 10.1002/nme.1620362105.

[41]

G. ZhaoX. Yu and R. Zhang, The new numerical method for solving the system of two-dimensional Burgers' equations, Comput. Math. Appl., 62 (2011), 3279-3291.  doi: 10.1016/j.camwa.2011.08.044.

show all references

References:
[1]

E. N. Aksan, A numerical solution of Burgers' equation by finite element method constructed on the method of discretization in time, Appl. Math. Comput., 170 (2005), 895-904.  doi: 10.1016/j.amc.2004.12.027.

[2]

R. Alexande, Diagonally Implicit Runge-Kutta Methods for Stiff O.D.E.'s, SIAM J. Numer. Anal., 14 (1977), 1006-1021.  doi: 10.1137/0714068.

[3]

A. AliG. Gardner and L. Gardner, A collocation solution for Burgers' equation using cubic B-spline finite elements, Comput. Method. Appl. M., 100 (1992), 325-337.  doi: 10.1016/0045-7825(92)90088-2.

[4]

P. Arminjon and C. Beauchamp, A finite element method for Burgers' equation in hydrodynamics, Int. J. num. Meth. Eng, 12 (1978), 415-428.  doi: 10.1002/nme.1620120304.

[5]

P. Arminjon and C. Beauchamp, Continuous and discontinuous finite element methods for Burgers' equation, Comput. Methods Appl. M., 25 (1981), 65-84.  doi: 10.1016/0045-7825(81)90069-4.

[6]

S. Brenner and L. Scott, The Mathematical Theory of Finite Element Methods, 3$^{rd}$ edition, Springer, New York, 2008. doi: 10.1007/978-0-387-75934-0.

[7]

J. Burgers, A mathematical model illustrating the theory of turbulence, Adv. Appl. Mech., 1 (1948), 171-199.  doi: 10.1016/S0065-2156(08)70100-5.

[8]

J. Burgers, Mathematical examples illustrating relations occuring in the theory of turbulent fluid motion, Springer, Dordrecht, 1995. doi: 10.1007/978-94-011-0195-0_10.

[9]

E. Burman, Error estimates for forward Euler shock capturing finite element approximations of the one-dimensional Burgers' equation, Math. Mod. Meth. Appl. S., 25 (2015), 2015-2042.  doi: 10.1142/S0218202515500517.

[10]

J. CaldwellP. Wanless and A. Cook, A finite element approach to Burgers' equation, Appl. Math. Model., 5 (1981), 189-193.  doi: 10.1016/0307-904X(81)90043-3.

[11]

G. Chen and X. Xie, A robust weak Galerkin finite element method for linear elasticity with strong symmetric stresses, Comput. Meth. Appl. Mat., 16 (2016), 389-408.  doi: 10.1515/cmam-2016-0012.

[12]

H. Chen and Z. Jiang, A characteristics-mixed finite element method for Burgers' equation, J. Appl. Math. Comput., 15 (2004), 29-51.  doi: 10.1007/BF02935745.

[13]

H. ChenP. Lu and X. Xu, A robust multilevel method for hybridizable discontinuous Galerkin method for the Helmholtz equation, J. Comput. Phys., 264 (2014), 133-151.  doi: 10.1016/j.jcp.2014.01.042.

[14]

Y. Chen and T. Zhang, A weak Galerkin finite element method for Burgers' equation, J. Comput. Appl. Math., 348 (2019), 103-109.  doi: 10.1016/j.cam.2018.08.044.

[15]

B. CockburnJ. Gopalakrishnan and N. Nguyen, Analysis of HDG methods for Stokes flow, Math. Comput., 80 (2011), 723-760.  doi: 10.1090/S0025-5718-2010-02410-X.

[16]

A. Dogan, A Galerkin finite element approach to Burgers' equation, Appl. Math. Comput., 157 (2004), 331-346.  doi: 10.1016/j.amc.2003.08.037.

[17]

R. Guzzi and L. Stefanutti, The Role of Airflow in Airborne Transmission of COVID 19, Int. J. Biol. Sci., 4 (2021), 121-131.  doi: 10.13133/2532-5876/17224.

[18]

Y. Han, H. Chen, X. Wang and X. Xie, EXtended HDG methods for second order elliptic interface problems, J. Sci. Comput., 84 (2020), 22. doi: 10.1007/s10915-020-01272-3.

[19]

J. Heywood and R. Rannacher, Finite-Element Approximation of the Nonstationary Navier-Stokes Problem. Part IV: Error Analysis for Second-Order Time Discretization, SIAM J. Numer. Anal., 27 (1990), 353-384.  doi: 10.1137/0727022.

[20]

X. HuP. Huang and X. Feng, Two-Grid Method for Burgers' Equation by a New Mixed Finite Element Scheme, Math. Model. Anal., 19 (2014), 1-17.  doi: 10.3846/13926292.2014.892902.

[21]

X. HuP. Huang and X. Feng, A new mixed finite element method based on the Crank-Nicolson scheme for Burgers' equation, Appl. Math-Czech., 61 (2016), 27-45.  doi: 10.1007/s10492-016-0120-3.

[22]

A. Hussein and H. Kashkool, Weak Galerkin finite element method for solving one-dimensional coupled Burgers' equations, J. Appl. Math. Comput., 63 (2020), 265-293.  doi: 10.1007/s12190-020-01317-8.

[23]

O. Karakashian and F. Pascal, Convergence of adaptive discontinuous Galerkin approximations of second order elliptic problems, SIAM J. Numer. Anal., 45 (2007), 641-665.  doi: 10.1137/05063979X.

[24]

S. KutluayA. Esen and I. Dag, Numerical solutions of the Burgers' equation by the least-squares quadratic B-spline finite element method, J. Comput. Appl. Math., 167 (2004), 21-33.  doi: 10.1016/j.cam.2003.09.043.

[25]

B. Li and X. Xie, Analysis of a family of HDG methods for second order elliptic problems, J. Comput. Appl. Math., 307 (2016), 37-51.  doi: 10.1016/j.cam.2016.04.027.

[26]

C. W. Lucchi, Improvement of MacCormack's scheme for Burgers' equation. Using a finite element method, Int. J. num. Meth. Eng, 15 (1980), 537-555.  doi: 10.1002/nme.1620150406.

[27]

R. Mittal and A. Tripathi, Numerical solutions of two-dimensional Burgers' equations using modified Bi-cubic B-spline finite elements, Eng. Comput., 32 (2015), 1275-1306.  doi: 10.1108/EC-04-2014-0067.

[28]

N. NguyenJ. Peraire and B. Cockburn, An implicit high-order hybridizable discontinuous Galerkin method for linear convection-diffusion equations, J. Comput. Phys., 228 (2009), 3232-3254.  doi: 10.1016/j.jcp.2009.01.030.

[29]

N. NguyenJ. Peraire and B. Cockburn, An implicit high-order hybridizable discontinuous Galerkin method for nonlinear convection-diffusion equations, J. Comput. Phys., 228 (2009), 8841-8855.  doi: 10.1016/j.jcp.2009.08.030.

[30]

T. ÖzişE. Aksan and A. Özdeş, A finite element approach for solution of Burgers' equation, Appl. Math. Comput., 139 (2003), 417-428.  doi: 10.1016/S0096-3003(02)00204-7.

[31]

A. PanyN. Nataraj and S. Singh, A new mixed finite element method for Burgers' equation, J. Appl. Math. Comput., 23 (2007), 43-55.  doi: 10.1007/BF02831957.

[32]

W. QiuJ. Shen and K. Shi, An HDG method for linear elasticity with strong symmetric stresses, Math. Comput., 87 (2016), 69-93.  doi: 10.1090/mcom/3249.

[33]

L. ShaoX. Feng and Y. He, The local discontinuous Galerkin finite element method for Burger's equation, Math. Comput. Model., 54 (2011), 2943-2954.  doi: 10.1016/j.mcm.2011.07.016.

[34]

D. ShiJ. Zhou and D. Shi, A new low order least squares nonconforming characteristics mixed finite element method for Burgers' equation, Appl. Math. Comput., 219 (2013), 11302-11310.  doi: 10.1016/j.amc.2013.05.037.

[35] Z. Shi and M. Wang, Finite Element Methods, Science Press, Beijing, 2013. 
[36]

H. SterckT. ManteuffelS. Mccormick and L. Olson, Numerical conservation properities of H(div)-conforming least-squares finite element methods for the Burgers equation, SIAM J. Sci. Comput., 26 (2005), 1573-1597.  doi: 10.1137/S1064827503430758.

[37]

R. Temam, Infinite-Dimensional Dynamical System in Mechanics and Physics, 2$^{nd}$ edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4684-0313-8.

[38]

Y. UçarN. Yaǧmurlu and İ. Çelikkaya, Operator splitting for numerical solution of the modified Burgers' equation using finite element method, Numer. Meth. Part. D. E., 35 (2019), 478-492.  doi: 10.1002/num.22309.

[39]

J. Warga, Optimal Control of Differential and Functional Equations, 1st edition, Academic Press, New York, 1972. doi: 10.1016/C2013-0-11669-8.

[40]

D. Winterscheidt and K. Surana, p-version least-squares finite element formulation of Burgers' equation, Int. J. num. Meth. Eng, 36 (2010), 3629-3646.  doi: 10.1002/nme.1620362105.

[41]

G. ZhaoX. Yu and R. Zhang, The new numerical method for solving the system of two-dimensional Burgers' equations, Comput. Math. Appl., 62 (2011), 3279-3291.  doi: 10.1016/j.camwa.2011.08.044.

Figure 1.  The domain : $ 4\times 4 $ (left) and $ 8\times 8 $ (right) meshes
Figure 2.  The domain: $ 2\times 2\times 2 $(left) and $ 4\times 4\times 4 $(right) meshes
Table 1.  History of convergence for Example 5.1 with $ \nu = 1, k = 1 $
(a) Method: HDG-I(l = 1)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 4\times 4 $ 2.1597e-01 2.9311e-01
$ 8\times 8 $ 5.4132e-02 2.00 1.4864e-01 0.98
$ 16\times 16 $ 1.3543e-02 2.00 7.4578e-02 0.99
$ 32\times 32 $ 3.3865e-03 2.00 3.7322e-02 1.00
$ 64\times 64 $ 8.4665e-04 2.00 1.8665e-02 1.00
(b) Method: HDG-II(l = 0)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 4\times 4 $2.4145e-01-3.1279e-01-
$ 8\times 8 $6.0180e-022.001.5806e-010.98
$ 16\times 16 $1.5038e-022.007.9255e-021.00
$32\times 32$3.7593e-032.003.9656e-021.00
$64\times 64$9.3980e-042.001.9832e-021.00
(a) Method: HDG-I(l = 1)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 4\times 4 $ 2.1597e-01 2.9311e-01
$ 8\times 8 $ 5.4132e-02 2.00 1.4864e-01 0.98
$ 16\times 16 $ 1.3543e-02 2.00 7.4578e-02 0.99
$ 32\times 32 $ 3.3865e-03 2.00 3.7322e-02 1.00
$ 64\times 64 $ 8.4665e-04 2.00 1.8665e-02 1.00
(b) Method: HDG-II(l = 0)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 4\times 4 $2.4145e-01-3.1279e-01-
$ 8\times 8 $6.0180e-022.001.5806e-010.98
$ 16\times 16 $1.5038e-022.007.9255e-021.00
$32\times 32$3.7593e-032.003.9656e-021.00
$64\times 64$9.3980e-042.001.9832e-021.00
Table 2.  History of convergence for Example 5.1 with $ \nu = 0.01, k = 1 $
(a) Method: HDG-I(l = 1)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \boldsymbol{q}(T)- \boldsymbol{q}_{h}(T)\rVert_0}{\lVert \boldsymbol{q}(T)\rVert_0} $
error order error order
$ 4\times 4 $ 1.1207e-01 - 2.9063e-01 -
$ 8\times 8 $ 3.3444e-02 1.74 1.4978e-01 0.96
$ 16\times 16 $ 8.5650e-03 1.97 7.4851e-02 1.00
$ 32\times 32 $ 2.1460e-03 2.00 3.7359e-02 1.00
$ 64\times 64 $ 5.3674e-04 2.00 1.8669e-02 1.00
(b) Method: HDG-II(l = 0)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \boldsymbol{q}(T)- \boldsymbol{q}_{h}(T)\rVert_0}{\lVert \boldsymbol{q}(T)\rVert_0} $
error order error order
$ 4\times 4 $1.3157e-01-3.0275e-01-
$ 8\times 8 $4.1890e-021.651.6350e-010.89
$ 16\times 16 $1.0394e-022.018.0699e-021.02
$32\times 32$2.5616e-032.023.9968e-021.01
$64\times 64$6.3806e-042.011.9936e-021.00
(a) Method: HDG-I(l = 1)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \boldsymbol{q}(T)- \boldsymbol{q}_{h}(T)\rVert_0}{\lVert \boldsymbol{q}(T)\rVert_0} $
error order error order
$ 4\times 4 $ 1.1207e-01 - 2.9063e-01 -
$ 8\times 8 $ 3.3444e-02 1.74 1.4978e-01 0.96
$ 16\times 16 $ 8.5650e-03 1.97 7.4851e-02 1.00
$ 32\times 32 $ 2.1460e-03 2.00 3.7359e-02 1.00
$ 64\times 64 $ 5.3674e-04 2.00 1.8669e-02 1.00
(b) Method: HDG-II(l = 0)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \boldsymbol{q}(T)- \boldsymbol{q}_{h}(T)\rVert_0}{\lVert \boldsymbol{q}(T)\rVert_0} $
error order error order
$ 4\times 4 $1.3157e-01-3.0275e-01-
$ 8\times 8 $4.1890e-021.651.6350e-010.89
$ 16\times 16 $1.0394e-022.018.0699e-021.02
$32\times 32$2.5616e-032.023.9968e-021.01
$64\times 64$6.3806e-042.011.9936e-021.00
Table 3.  History of convergence for Example 5.1 with $ \nu = 1, k = 2 $
(a) Method: HDG-I(l = 2)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \boldsymbol{q}(T)- \boldsymbol{q}_{h}(T)\rVert_0}{\lVert \boldsymbol{q}(T)\rVert_0} $
error order error order
$ 4\times 4 $ 1.3700e-02 - 4.1763e-02 -
$ 8\times 8 $ 1.5937e-03 3.10 1.0597e-02 1.98
$ 16\times 16 $ 1.9284e-04 3.05 2.6642e-03 1.99
$ 32\times 32 $ 2.3736e-05 3.02 6.6756e-04 2.00
$ 64\times 64 $ 2.9449e-06 3.01 1.6705e-04 2.00
(b) Method: HDG-II(l = 1)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \boldsymbol{q}(T)- \boldsymbol{q}_{h}(T)\rVert_0}{\lVert \boldsymbol{q}(T)\rVert_0} $
error order error order
$ 4\times 4 $1.4714e-02-4.3007e-02-
$ 8\times 8 $1.7267e-033.091.0913e-021.98
$ 16\times 16 $2.1016e-043.042.7449e-031.99
$ 32\times 32 $2.5953e-053.026.8800e-042.00
$ 64\times 64 $3.2256e-063.011.7220e-042.00
(a) Method: HDG-I(l = 2)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \boldsymbol{q}(T)- \boldsymbol{q}_{h}(T)\rVert_0}{\lVert \boldsymbol{q}(T)\rVert_0} $
error order error order
$ 4\times 4 $ 1.3700e-02 - 4.1763e-02 -
$ 8\times 8 $ 1.5937e-03 3.10 1.0597e-02 1.98
$ 16\times 16 $ 1.9284e-04 3.05 2.6642e-03 1.99
$ 32\times 32 $ 2.3736e-05 3.02 6.6756e-04 2.00
$ 64\times 64 $ 2.9449e-06 3.01 1.6705e-04 2.00
(b) Method: HDG-II(l = 1)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \boldsymbol{q}(T)- \boldsymbol{q}_{h}(T)\rVert_0}{\lVert \boldsymbol{q}(T)\rVert_0} $
error order error order
$ 4\times 4 $1.4714e-02-4.3007e-02-
$ 8\times 8 $1.7267e-033.091.0913e-021.98
$ 16\times 16 $2.1016e-043.042.7449e-031.99
$ 32\times 32 $2.5953e-053.026.8800e-042.00
$ 64\times 64 $3.2256e-063.011.7220e-042.00
Table 4.  History of convergence for Example 5.1 with $ \nu = 0.01, k = 2 $
(a) Method: HDG-I(l = 2)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 4\times 4 $ 9.4294e-02 1.0218e-01
$ 8\times 8 $ 1.2210e-02 2.95 1.6179e-02 2.66
$ 16\times 16 $ 1.5272e-03 3.00 3.0797e-03 2.39
$ 32\times 32 $ 1.9068e-04 3.00 6.9507e-04 2.15
$ 64\times 64 $ 2.3817e-05 3.00 1.6880e-04 2.04
(b) Method: HDG-II(l = 1)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 4\times 4 $9.4922e-021.0298e-01
$ 8\times 8 $1.2259e-022.951.6577e-022.64
$ 16\times 16 $1.5310e-033.003.1770e-032.38
$ 32\times 32 $1.9114e-043.007.2041e-042.14
$ 64\times 64 $2.3877e-053.001.7525e-042.04
(a) Method: HDG-I(l = 2)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 4\times 4 $ 9.4294e-02 1.0218e-01
$ 8\times 8 $ 1.2210e-02 2.95 1.6179e-02 2.66
$ 16\times 16 $ 1.5272e-03 3.00 3.0797e-03 2.39
$ 32\times 32 $ 1.9068e-04 3.00 6.9507e-04 2.15
$ 64\times 64 $ 2.3817e-05 3.00 1.6880e-04 2.04
(b) Method: HDG-II(l = 1)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 4\times 4 $9.4922e-021.0298e-01
$ 8\times 8 $1.2259e-022.951.6577e-022.64
$ 16\times 16 $1.5310e-033.003.1770e-032.38
$ 32\times 32 $1.9114e-043.007.2041e-042.14
$ 64\times 64 $2.3877e-053.001.7525e-042.04
Table 5.  History of convergence with $ k = 3 $: Example 5.2
$ \Delta t $ HDG-I$ (l=3) $ HDG-II$ (l=2) $
$ \frac{\lVert \mathit{\boldsymbol{u}}(T)-\mathit{\boldsymbol{u}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{u}}(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{u}}(T)-\mathit{\boldsymbol{u}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{u}}(T)\rVert_0} $
error order error order
0.2 2.2145e-03 2.2145e-03
0.1 3.7353e-04 2.57 3.7353e-04 2.57
0.05 5.7074e-05 2.71 5.7074e-05 2.71
0.025 8.1013e-06 2.82 8.1013e-06 2.82
0.0125 1.0947e-06 2.89 1.0947e-06 2.89
$ \Delta t $ HDG-I$ (l=3) $ HDG-II$ (l=2) $
$ \frac{\lVert \mathit{\boldsymbol{u}}(T)-\mathit{\boldsymbol{u}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{u}}(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{u}}(T)-\mathit{\boldsymbol{u}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{u}}(T)\rVert_0} $
error order error order
0.2 2.2145e-03 2.2145e-03
0.1 3.7353e-04 2.57 3.7353e-04 2.57
0.05 5.7074e-05 2.71 5.7074e-05 2.71
0.025 8.1013e-06 2.82 8.1013e-06 2.82
0.0125 1.0947e-06 2.89 1.0947e-06 2.89
Table 6.  History of convergence with $ k = 1 $: Example 5.2
(a) Method: HDG-I(l = 1)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 8\times 8 $ 1.5445e-01 4.2222e-01
$ 16\times 16 $ 4.2801e-02 1.85 1.9709e-01 1.10
$ 32\times 32 $ 1.0908e-02 1.97 9.8335e-02 1.00
$ 64\times 64 $ 2.7412e-03 1.99 4.9190e-02 1.00
$ 128\times 128 $ 6.8606e-04 2.00 2.4598e-02 1.00
(b) Method: HDG-II(l = 0)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 8\times 8 $1.7900e-014.5177e-01
$ 16\times 16 $4.7423e-021.922.0942e-011.11
$ 32\times 32 $1.1971e-021.991.0380e-011.01
$ 64\times 64 $3.0019e-032.005.1851e-021.00
$ 128\times 128 $7.5102e-042.002.5920e-021.00
(a) Method: HDG-I(l = 1)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 8\times 8 $ 1.5445e-01 4.2222e-01
$ 16\times 16 $ 4.2801e-02 1.85 1.9709e-01 1.10
$ 32\times 32 $ 1.0908e-02 1.97 9.8335e-02 1.00
$ 64\times 64 $ 2.7412e-03 1.99 4.9190e-02 1.00
$ 128\times 128 $ 6.8606e-04 2.00 2.4598e-02 1.00
(b) Method: HDG-II(l = 0)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 8\times 8 $1.7900e-014.5177e-01
$ 16\times 16 $4.7423e-021.922.0942e-011.11
$ 32\times 32 $1.1971e-021.991.0380e-011.01
$ 64\times 64 $3.0019e-032.005.1851e-021.00
$ 128\times 128 $7.5102e-042.002.5920e-021.00
Table 7.  History of convergence with $ k = 2 $: Example 5.2
(a) Method: HDG-I(l = 2)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 8\times 8 $ 2.8115e-02 4.9545e-02
$ 16\times 16 $ 4.7515e-03 2.56 2.1126e-02 1.23
$ 32\times 32 $ 6.1910e-04 2.94 5.6085e-03 1.91
$ 64\times 64 $ 7.7864e-05 2.99 1.4177e-03 1.98
$ 128\times 128 $ 9.7479e-06 3.00 3.5571e-04 1.99
(b) Method: HDG-II(l = 1)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 8\times 8 $2.9126e-025.0053e-02
$ 16\times 16 $4.8563e-032.582.1191e-021.24
$ 32\times 32 $6.3425e-042.945.6117e-031.92
$ 64\times 64 $7.9961e-052.991.4199e-031.98
$ 128\times 128 $1.0021e-053.003.5663e-041.99
(a) Method: HDG-I(l = 2)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 8\times 8 $ 2.8115e-02 4.9545e-02
$ 16\times 16 $ 4.7515e-03 2.56 2.1126e-02 1.23
$ 32\times 32 $ 6.1910e-04 2.94 5.6085e-03 1.91
$ 64\times 64 $ 7.7864e-05 2.99 1.4177e-03 1.98
$ 128\times 128 $ 9.7479e-06 3.00 3.5571e-04 1.99
(b) Method: HDG-II(l = 1)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 8\times 8 $2.9126e-025.0053e-02
$ 16\times 16 $4.8563e-032.582.1191e-021.24
$ 32\times 32 $6.3425e-042.945.6117e-031.92
$ 64\times 64 $7.9961e-052.991.4199e-031.98
$ 128\times 128 $1.0021e-053.003.5663e-041.99
Table 8.  History of convergence with $ k = 1 $: Example 5.3
(a) Method: HDG-I(l = 1)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 2\times 2\times 2 $ 6.9815e-01 5.6066e-01
$ 4\times 4\times 4 $ 1.7672e-01 1.98 3.0285e-01 0.89
$ 8\times 8\times 8 $ 4.4207e-02 2.00 1.5438e-01 0.97
$ 16\times 16\times 16 $ 1.1050e-02 2.00 7.7566e-02 0.99
$ 32\times 32\times 32 $ 2.7621e-03 2.00 3.8830e-02 1.00
(b) Method: HDG-II(l = 0)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 2\times 2\times 2 $8.8064e-016.1105e-01
$ 4\times 4\times 4 $2.0917e-012.073.1971e-010.93
$ 8\times 8\times 8 $5.1528e-022.021.6186e-010.98
$ 16\times 16\times 16 $1.2835e-022.008.1193e-020.99
$ 32\times 32\times 32 $3.2055e-032.004.0630e-021.00
(a) Method: HDG-I(l = 1)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 2\times 2\times 2 $ 6.9815e-01 5.6066e-01
$ 4\times 4\times 4 $ 1.7672e-01 1.98 3.0285e-01 0.89
$ 8\times 8\times 8 $ 4.4207e-02 2.00 1.5438e-01 0.97
$ 16\times 16\times 16 $ 1.1050e-02 2.00 7.7566e-02 0.99
$ 32\times 32\times 32 $ 2.7621e-03 2.00 3.8830e-02 1.00
(b) Method: HDG-II(l = 0)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 2\times 2\times 2 $8.8064e-016.1105e-01
$ 4\times 4\times 4 $2.0917e-012.073.1971e-010.93
$ 8\times 8\times 8 $5.1528e-022.021.6186e-010.98
$ 16\times 16\times 16 $1.2835e-022.008.1193e-020.99
$ 32\times 32\times 32 $3.2055e-032.004.0630e-021.00
Table 9.  History of convergence with $ k = 2 $: Example 5.3
(a) Method: HDG-I(l = 2)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 2\times 2\times 2 $ 1.3095e-01 1.9090e-01
$ 4\times 4\times 4 $ 1.4825e-02 3.14 5.3052e-02 1.85
$ 8\times 8\times 8 $ 1.7531e-03 3.08 1.3659e-02 1.96
$ 16\times 16\times 16 $ 2.1463e-04 3.03 3.4459e-03 1.99
(b) Method: HDG-II(l = 1)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 2\times 2\times 2 $1.4705e-011.9582e-01
$ 4\times 4\times 4 $1.6034e-023.205.3962e-021.86
$ 8\times 8\times 8 $1.8868e-033.091.3871e-021.96
$ 16\times 16\times 16 $2.3104e-043.033.4979e-031.99
$ 32\times 32\times 32 $2.8681e-053.018.7713e-042.00
(a) Method: HDG-I(l = 2)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 2\times 2\times 2 $ 1.3095e-01 1.9090e-01
$ 4\times 4\times 4 $ 1.4825e-02 3.14 5.3052e-02 1.85
$ 8\times 8\times 8 $ 1.7531e-03 3.08 1.3659e-02 1.96
$ 16\times 16\times 16 $ 2.1463e-04 3.03 3.4459e-03 1.99
(b) Method: HDG-II(l = 1)
mesh $ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $ $ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
error order error order
$ 2\times 2\times 2 $1.4705e-011.9582e-01
$ 4\times 4\times 4 $1.6034e-023.205.3962e-021.86
$ 8\times 8\times 8 $1.8868e-033.091.3871e-021.96
$ 16\times 16\times 16 $2.3104e-043.033.4979e-031.99
$ 32\times 32\times 32 $2.8681e-053.018.7713e-042.00
[1]

Arnaud Debussche, Jacques Printems. Convergence of a semi-discrete scheme for the stochastic Korteweg-de Vries equation. Discrete and Continuous Dynamical Systems - B, 2006, 6 (4) : 761-781. doi: 10.3934/dcdsb.2006.6.761

[2]

Sylvie Benzoni-Gavage, Pierre Huot. Existence of semi-discrete shocks. Discrete and Continuous Dynamical Systems, 2002, 8 (1) : 163-190. doi: 10.3934/dcds.2002.8.163

[3]

Mahboub Baccouch. Superconvergence of the semi-discrete local discontinuous Galerkin method for nonlinear KdV-type problems. Discrete and Continuous Dynamical Systems - B, 2019, 24 (1) : 19-54. doi: 10.3934/dcdsb.2018104

[4]

Maike Schulte, Anton Arnold. Discrete transparent boundary conditions for the Schrodinger equation -- a compact higher order scheme. Kinetic and Related Models, 2008, 1 (1) : 101-125. doi: 10.3934/krm.2008.1.101

[5]

Yingwen Guo, Yinnian He. Fully discrete finite element method based on second-order Crank-Nicolson/Adams-Bashforth scheme for the equations of motion of Oldroyd fluids of order one. Discrete and Continuous Dynamical Systems - B, 2015, 20 (8) : 2583-2609. doi: 10.3934/dcdsb.2015.20.2583

[6]

Horst Osberger. Long-time behavior of a fully discrete Lagrangian scheme for a family of fourth order equations. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 405-434. doi: 10.3934/dcds.2017017

[7]

Enrique Zuazua. Controllability of partial differential equations and its semi-discrete approximations. Discrete and Continuous Dynamical Systems, 2002, 8 (2) : 469-513. doi: 10.3934/dcds.2002.8.469

[8]

Fabio Camilli, Francisco Silva. A semi-discrete approximation for a first order mean field game problem. Networks and Heterogeneous Media, 2012, 7 (2) : 263-277. doi: 10.3934/nhm.2012.7.263

[9]

Carlo Bardaro, Ilaria Mantellini. Boundedness properties of semi-discrete sampling operators in Mellin–Lebesgue spaces. Mathematical Foundations of Computing, 2022, 5 (3) : 219-229. doi: 10.3934/mfc.2021031

[10]

Vladislav Balashov, Alexander Zlotnik. An energy dissipative semi-discrete finite-difference method on staggered meshes for the 3D compressible isothermal Navier–Stokes–Cahn–Hilliard equations. Journal of Computational Dynamics, 2020, 7 (2) : 291-312. doi: 10.3934/jcd.2020012

[11]

Thuy N. T. Nguyen. Carleman estimates for semi-discrete parabolic operators with a discontinuous diffusion coefficient and applications to controllability. Mathematical Control and Related Fields, 2014, 4 (2) : 203-259. doi: 10.3934/mcrf.2014.4.203

[12]

Feng Bao, Yanzhao Cao, Weidong Zhao. A first order semi-discrete algorithm for backward doubly stochastic differential equations. Discrete and Continuous Dynamical Systems - B, 2015, 20 (5) : 1297-1313. doi: 10.3934/dcdsb.2015.20.1297

[13]

Ansgar Jüngel, Oliver Leingang. Blow-up of solutions to semi-discrete parabolic-elliptic Keller-Segel models. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 4755-4782. doi: 10.3934/dcdsb.2019029

[14]

Damien Allonsius, Franck Boyer. Boundary null-controllability of semi-discrete coupled parabolic systems in some multi-dimensional geometries. Mathematical Control and Related Fields, 2020, 10 (2) : 217-256. doi: 10.3934/mcrf.2019037

[15]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 4907-4926. doi: 10.3934/dcdsb.2020319

[16]

Matúš Tibenský, Angela Handlovičová. Convergence analysis of the discrete duality finite volume scheme for the regularised Heston model. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1181-1195. doi: 10.3934/dcdss.2020226

[17]

Xiaojie Wang. Weak error estimates of the exponential Euler scheme for semi-linear SPDEs without Malliavin calculus. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 481-497. doi: 10.3934/dcds.2016.36.481

[18]

Kun Wang, Yinnian He, Yueqiang Shang. Fully discrete finite element method for the viscoelastic fluid motion equations. Discrete and Continuous Dynamical Systems - B, 2010, 13 (3) : 665-684. doi: 10.3934/dcdsb.2010.13.665

[19]

Abdeladim El Akri, Lahcen Maniar. Uniform indirect boundary controllability of semi-discrete $ 1 $-$ d $ coupled wave equations. Mathematical Control and Related Fields, 2020, 10 (4) : 669-698. doi: 10.3934/mcrf.2020015

[20]

John D. Towers. The Lax-Friedrichs scheme for interaction between the inviscid Burgers equation and multiple particles. Networks and Heterogeneous Media, 2020, 15 (1) : 143-169. doi: 10.3934/nhm.2020007

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (321)
  • HTML views (408)
  • Cited by (0)

Other articles
by authors

[Back to Top]