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August  2019, 12(4): 885-908. doi: 10.3934/krm.2019033

An entropy stable high-order discontinuous Galerkin method for cross-diffusion gradient flow systems

1. 

Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA

2. 

Department of Mathematics, Imperial College London, London SW7 2AZ, UK

3. 

Division of Applied Mathematics, Brown University, Providence, RI 02912, USA

* Corresponding author: José A. Carrillo

Received  October 2018 Revised  February 2019 Published  May 2019

Fund Project: JAC was partially supported by the EPSRC grant number EP/P031587/1. JAC would like to thank the Department of Applied Mathematics at Brown University for their kind hospitality and for the support through the IBM Visiting Professorship scheme. CWS was partially supported by ARO grant W911NF-16-1-0103 and NSF grant DMS-1719410

As an extension of our previous work in [41], we develop a discontinuous Galerkin method for solving cross-diffusion systems with a formal gradient flow structure. These systems are associated with non-increasing entropy functionals. For a class of problems, the positivity (non-negativity) of solutions is also expected, which is implied by the physical model and is crucial to the entropy structure. The semi-discrete numerical scheme we propose is entropy stable. Furthermore, the scheme is also compatible with the positivity-preserving procedure in [43] in many scenarios, hence the resulting fully discrete scheme is able to produce non-negative solutions. The method can be applied to both one-dimensional problems and two-dimensional problems on Cartesian meshes. Numerical examples are given to examine the performance of the method.

Citation: Zheng Sun, José A. Carrillo, Chi-Wang Shu. An entropy stable high-order discontinuous Galerkin method for cross-diffusion gradient flow systems. Kinetic & Related Models, 2019, 12 (4) : 885-908. doi: 10.3934/krm.2019033
References:
[1]

F. Bassi and S. Rebay, A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier–Stokes equations, Journal of Computational Physics, 131 (1997), 267-279. doi: 10.1006/jcph.1996.5572. Google Scholar

[2]

J. BerendsenM. Burger and J.-F. Pietschmann, On a cross-diffusion model for multiple species with nonlocal interaction and size exclusion, Nonlinear Analysis, 159 (2017), 10-39. doi: 10.1016/j.na.2017.03.010. Google Scholar

[3]

M. Bessemoulin-Chatard and F. Filbet, A finite volume scheme for nonlinear degenerate parabolic equations, SIAM Journal on Scientific Computing, 34 (2012), B559–B583. doi: 10.1137/110853807. Google Scholar

[4]

M. BrunaM. BurgerH. Ranetbauer and M.-T. Wolfram, Cross-diffusion systems with excluded-volume effects and asymptotic gradient flow structures, Journal of Nonlinear Science, 27 (2017), 687-719. doi: 10.1007/s00332-016-9348-z. Google Scholar

[5]

M. BurgerJ. A. Carrillo and M.-T. Wolfram, A mixed finite element method for nonlinear diffusion equations, Kinetic and Related Models, 3 (2010), 59-83. doi: 10.3934/krm.2010.3.59. Google Scholar

[6]

J. A. CarrilloA. Chertock and Y. Huang, A finite-volume method for nonlinear nonlocal equations with a gradient flow structure, Communications in Computational Physics, 17 (2015), 233-258. doi: 10.4208/cicp.160214.010814a. Google Scholar

[7]

J. A. Carrillo, K. Craig and F. S. Patacchini, A blob method for diffusion, Calc. Var. Partial Differential Equations, 58 (2019), Art. 53, 53 pp, arXiv: 1709.09195. doi: 10.1007/s00526-019-1486-3. Google Scholar

[8]

J. A. Carrillo, F. Filbet and M. Schmidtchen, Convergence of a finite volume scheme for a system of interacting species with cross-diffusion, preprint, arXiv: 1804.04385.Google Scholar

[9]

J. A. CarrilloY. HuangF. S. Patacchini and G. Wolansky, Numerical study of a particle method for gradient flows, Kinetic and Related Models, 10 (2017), 613-641. doi: 10.3934/krm.2017025. Google Scholar

[10]

J. A. CarrilloB. DüringD. Matthes and D. McCormick, A Lagrangian scheme for the solution of nonlinear diffusion equations using moving simplex meshes, Journal of Scientific Computing, 75 (2018), 1463-1499. doi: 10.1007/s10915-017-0594-5. Google Scholar

[11]

J. A. Carrillo and J. S. Moll, Numerical simulation of diffusive and aggregation phenomena in nonlinear continuity equations by evolving diffeomorphisms, SIAM Journal on Scientific Computing, 31 (2009/10), 4305-4329. doi: 10.1137/080739574. Google Scholar

[12]

J. A. CarrilloH. Ranetbauer and M.-T. Wolfram, Numerical simulation of nonlinear continuity equations by evolving diffeomorphisms, Journal of Computational Physics, 327 (2016), 186-202. doi: 10.1016/j.jcp.2016.09.040. Google Scholar

[13]

T. Chen and C.-W. Shu, Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws, Journal of Computational Physics, 345 (2017), 427-461. doi: 10.1016/j.jcp.2017.05.025. Google Scholar

[14]

Y. Cheng and C.-W. Shu, A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives, Mathematics of Computation, 77 (2008), 699-730. doi: 10.1090/S0025-5718-07-02045-5. Google Scholar

[15]

B. CockburnS. Hou and C.-W. Shu, The Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws Ⅳ: The multidimensional case, Mathematics of Computation, 54 (1990), 545-581. doi: 10.2307/2008501. Google Scholar

[16]

B. CockburnS.-Y. Lin and C.-W. Shu, TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws Ⅲ: one-dimensional systems, Journal of Computational Physics, 84 (1989), 90-113. doi: 10.1016/0021-9991(89)90183-6. Google Scholar

[17]

B. Cockburn and C.-W. Shu, TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws Ⅱ: General framework, Mathematics of Computation, 52 (1989), 411-435. doi: 10.2307/2008474. Google Scholar

[18]

B. Cockburn and C.-W. Shu, The Runge–Kutta local projection ${P}^1$-discontinuous-Galerkin finite element method for scalar conservation laws, ESAIM: Mathematical Modelling and Numerical Analysis, 25 (1991), 337-361. doi: 10.1051/m2an/1991250303371. Google Scholar

[19]

B. Cockburn and C.-W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM Journal on Numerical Analysis, 35 (1998), 2440-2463. doi: 10.1137/S0036142997316712. Google Scholar

[20]

B. Cockburn and C.-W. Shu, The Runge–Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, Journal of Computational Physics, 141 (1998), 199-224. doi: 10.1006/jcph.1998.5892. Google Scholar

[21]

K. Craig and A. Bertozzi, A blob method for the aggregation equation, Mathematics of Computation, 85 (2016), 1681-1717. doi: 10.1090/mcom3033. Google Scholar

[22]

J. EscherM. HillairetP. Laurencot and C. Walker, Global weak solutions for a degenerate parabolic system modeling the spreading of insoluble surfactant, Indiana University Mathematics Journal, 60 (2011), 1975-2019. doi: 10.1512/iumj.2011.60.4447. Google Scholar

[23]

S. Gottlieb, C.-W. Shu and E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Review, 43 (2001), 89–112. doi: 10.1137/S003614450036757X. Google Scholar

[24]

S. HittmeirH. RanetbauerC. Schmeiser and M.-T. Wolfram, Derivation and analysis of continuum models for crossing pedestrian traffic, Mathematical Models and Methods in Applied Sciences, 27 (2017), 1301-1325. doi: 10.1142/S0218202517400164. Google Scholar

[25]

T. L. Jackson and H. M. Byrne, A mechanical model of tumor encapsulation and transcapsular spread, Mathematical Biosciences, 180 (2002), 307-328. doi: 10.1016/S0025-5564(02)00118-9. Google Scholar

[26]

O. Jensen and J. Grotberg, Insoluble surfactant spreading on a thin viscous film: Shock evolution and film rupture, Journal of Fluid Mechanics, 240 (1992), 259-288. doi: 10.1017/S0022112092000090. Google Scholar

[27]

O. JungeD. Matthes and H. Osberger, A fully discrete variational scheme for solving nonlinear Fokker-Planck equations in multiple space dimensions, SIAM Journal on Numerical Analysis, 55 (2017), 419-443. doi: 10.1137/16M1056560. Google Scholar

[28]

A. Jüngel, The boundedness-by-entropy method for cross-diffusion systems, Nonlinearity, 28 (2015), 1963-2001. doi: 10.1088/0951-7715/28/6/1963. Google Scholar

[29]

A. Jüngel and I. V. Stelzer, Entropy structure of a cross-diffusion tumor-growth model, Mathematical Models and Methods in Applied Sciences, 22 (2012), 1250009, 26pp. doi: 10.1142/S0218202512500091. Google Scholar

[30]

A. Jüngel and N. Zamponi, Qualitative behavior of solutions to cross-diffusion systems from population dynamics, Journal of Mathematical Analysis and Applications, 440 (2016), 794-809. doi: 10.1016/j.jmaa.2016.03.076. Google Scholar

[31]

A. Jüngel and N. Zamponi, Analysis of degenerate cross-diffusion population models with volume filling, Annales de l'Institut Henri Poincaré C, Analyse Non Linéaire, 34 (2017), 1-29. doi: 10.1016/j.anihpc.2015.08.003. Google Scholar

[32]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, Journal of Theoretical Biology, 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5. Google Scholar

[33]

H. Liu and H. Yu, The entropy satisfying discontinuous Galerkin method for Fokker-Planck equations, Journal of Scientific Computing, 62 (2015), 803-830. doi: 10.1007/s10915-014-9878-1. Google Scholar

[34]

H. Liu and Z. Wang, An entropy satisfying discontinuous Galerkin method for nonlinear Fokker–Planck equations, Journal of Scientific Computing, 68 (2016), 1217-1240. doi: 10.1007/s10915-016-0174-0. Google Scholar

[35]

H. Liu and Z. Wang, A free energy satisfying discontinuous Galerkin method for one-dimensional Poisson–Nernst–Planck systems, Journal of Computational Physics, 328 (2017), 413-437. doi: 10.1016/j.jcp.2016.10.008. Google Scholar

[36]

H. Liu and J. Yan, The direct discontinuous Galerkin (DDG) methods for diffusion problems, SIAM Journal on Numerical Analysis, 47 (2009), 675-698. doi: 10.1137/080720255. Google Scholar

[37]

A. A. H. Oulhaj, A finite volume scheme for a seawater intrusion model with cross-diffusion, in Finite Volumes for Complex Applications VIII – Methods and Theoretical Aspects (eds. C. Cancès and P. Omnes), Springer International Publishing, 199 (2017), 421–429. Google Scholar

[38]

W. H. Reed and T. Hill, Triangular mesh methods for the neutron transport equation, Technical report, Los Alamos Scientific Lab., N. Mex.(USA), 1973.Google Scholar

[39]

N. ShigesadaK. Kawasaki and E. Teramoto, Spatial segregation of interacting species, Journal of Theoretical Biology, 79 (1979), 83-99. doi: 10.1016/0022-5193(79)90258-3. Google Scholar

[40]

S. SrinivasanaJ. Poggiea and X. Zhang, A positivity-preserving high order discontinuous Galerkin scheme for convection-diffusion equations, Journal of Computational Physics, 366 (2018), 120-143. doi: 10.1016/j.jcp.2018.04.002. Google Scholar

[41]

Z. SunJ. A. Carrillo and C.-W. Shu, A discontinuous Galerkin method for nonlinear parabolic equations and gradient flow problems with interaction potentials, Journal of Computational Physics, 352 (2018), 76-104. doi: 10.1016/j.jcp.2017.09.050. Google Scholar

[42]

Z. Sun, J. A. Carrillo and C.-W. Shu, An entropy stable high-order discontinuous Galerkin method for cross-diffusion gradient flow systems, preprint, arXiv: 1810.03221.Google Scholar

[43]

X. Zhang, On positivity-preserving high order discontinuous Galerkin schemes for compressible Navier–Stokes equations, Journal of Computational Physics, 328 (2017), 301-343. doi: 10.1016/j.jcp.2016.10.002. Google Scholar

[44]

X. Zhang and C.-W. Shu, On maximum-principle-satisfying high order schemes for scalar conservation laws, Journal of Computational Physics, 229 (2010), 3091-3120. doi: 10.1016/j.jcp.2009.12.030. Google Scholar

show all references

References:
[1]

F. Bassi and S. Rebay, A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier–Stokes equations, Journal of Computational Physics, 131 (1997), 267-279. doi: 10.1006/jcph.1996.5572. Google Scholar

[2]

J. BerendsenM. Burger and J.-F. Pietschmann, On a cross-diffusion model for multiple species with nonlocal interaction and size exclusion, Nonlinear Analysis, 159 (2017), 10-39. doi: 10.1016/j.na.2017.03.010. Google Scholar

[3]

M. Bessemoulin-Chatard and F. Filbet, A finite volume scheme for nonlinear degenerate parabolic equations, SIAM Journal on Scientific Computing, 34 (2012), B559–B583. doi: 10.1137/110853807. Google Scholar

[4]

M. BrunaM. BurgerH. Ranetbauer and M.-T. Wolfram, Cross-diffusion systems with excluded-volume effects and asymptotic gradient flow structures, Journal of Nonlinear Science, 27 (2017), 687-719. doi: 10.1007/s00332-016-9348-z. Google Scholar

[5]

M. BurgerJ. A. Carrillo and M.-T. Wolfram, A mixed finite element method for nonlinear diffusion equations, Kinetic and Related Models, 3 (2010), 59-83. doi: 10.3934/krm.2010.3.59. Google Scholar

[6]

J. A. CarrilloA. Chertock and Y. Huang, A finite-volume method for nonlinear nonlocal equations with a gradient flow structure, Communications in Computational Physics, 17 (2015), 233-258. doi: 10.4208/cicp.160214.010814a. Google Scholar

[7]

J. A. Carrillo, K. Craig and F. S. Patacchini, A blob method for diffusion, Calc. Var. Partial Differential Equations, 58 (2019), Art. 53, 53 pp, arXiv: 1709.09195. doi: 10.1007/s00526-019-1486-3. Google Scholar

[8]

J. A. Carrillo, F. Filbet and M. Schmidtchen, Convergence of a finite volume scheme for a system of interacting species with cross-diffusion, preprint, arXiv: 1804.04385.Google Scholar

[9]

J. A. CarrilloY. HuangF. S. Patacchini and G. Wolansky, Numerical study of a particle method for gradient flows, Kinetic and Related Models, 10 (2017), 613-641. doi: 10.3934/krm.2017025. Google Scholar

[10]

J. A. CarrilloB. DüringD. Matthes and D. McCormick, A Lagrangian scheme for the solution of nonlinear diffusion equations using moving simplex meshes, Journal of Scientific Computing, 75 (2018), 1463-1499. doi: 10.1007/s10915-017-0594-5. Google Scholar

[11]

J. A. Carrillo and J. S. Moll, Numerical simulation of diffusive and aggregation phenomena in nonlinear continuity equations by evolving diffeomorphisms, SIAM Journal on Scientific Computing, 31 (2009/10), 4305-4329. doi: 10.1137/080739574. Google Scholar

[12]

J. A. CarrilloH. Ranetbauer and M.-T. Wolfram, Numerical simulation of nonlinear continuity equations by evolving diffeomorphisms, Journal of Computational Physics, 327 (2016), 186-202. doi: 10.1016/j.jcp.2016.09.040. Google Scholar

[13]

T. Chen and C.-W. Shu, Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws, Journal of Computational Physics, 345 (2017), 427-461. doi: 10.1016/j.jcp.2017.05.025. Google Scholar

[14]

Y. Cheng and C.-W. Shu, A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives, Mathematics of Computation, 77 (2008), 699-730. doi: 10.1090/S0025-5718-07-02045-5. Google Scholar

[15]

B. CockburnS. Hou and C.-W. Shu, The Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws Ⅳ: The multidimensional case, Mathematics of Computation, 54 (1990), 545-581. doi: 10.2307/2008501. Google Scholar

[16]

B. CockburnS.-Y. Lin and C.-W. Shu, TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws Ⅲ: one-dimensional systems, Journal of Computational Physics, 84 (1989), 90-113. doi: 10.1016/0021-9991(89)90183-6. Google Scholar

[17]

B. Cockburn and C.-W. Shu, TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws Ⅱ: General framework, Mathematics of Computation, 52 (1989), 411-435. doi: 10.2307/2008474. Google Scholar

[18]

B. Cockburn and C.-W. Shu, The Runge–Kutta local projection ${P}^1$-discontinuous-Galerkin finite element method for scalar conservation laws, ESAIM: Mathematical Modelling and Numerical Analysis, 25 (1991), 337-361. doi: 10.1051/m2an/1991250303371. Google Scholar

[19]

B. Cockburn and C.-W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM Journal on Numerical Analysis, 35 (1998), 2440-2463. doi: 10.1137/S0036142997316712. Google Scholar

[20]

B. Cockburn and C.-W. Shu, The Runge–Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, Journal of Computational Physics, 141 (1998), 199-224. doi: 10.1006/jcph.1998.5892. Google Scholar

[21]

K. Craig and A. Bertozzi, A blob method for the aggregation equation, Mathematics of Computation, 85 (2016), 1681-1717. doi: 10.1090/mcom3033. Google Scholar

[22]

J. EscherM. HillairetP. Laurencot and C. Walker, Global weak solutions for a degenerate parabolic system modeling the spreading of insoluble surfactant, Indiana University Mathematics Journal, 60 (2011), 1975-2019. doi: 10.1512/iumj.2011.60.4447. Google Scholar

[23]

S. Gottlieb, C.-W. Shu and E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Review, 43 (2001), 89–112. doi: 10.1137/S003614450036757X. Google Scholar

[24]

S. HittmeirH. RanetbauerC. Schmeiser and M.-T. Wolfram, Derivation and analysis of continuum models for crossing pedestrian traffic, Mathematical Models and Methods in Applied Sciences, 27 (2017), 1301-1325. doi: 10.1142/S0218202517400164. Google Scholar

[25]

T. L. Jackson and H. M. Byrne, A mechanical model of tumor encapsulation and transcapsular spread, Mathematical Biosciences, 180 (2002), 307-328. doi: 10.1016/S0025-5564(02)00118-9. Google Scholar

[26]

O. Jensen and J. Grotberg, Insoluble surfactant spreading on a thin viscous film: Shock evolution and film rupture, Journal of Fluid Mechanics, 240 (1992), 259-288. doi: 10.1017/S0022112092000090. Google Scholar

[27]

O. JungeD. Matthes and H. Osberger, A fully discrete variational scheme for solving nonlinear Fokker-Planck equations in multiple space dimensions, SIAM Journal on Numerical Analysis, 55 (2017), 419-443. doi: 10.1137/16M1056560. Google Scholar

[28]

A. Jüngel, The boundedness-by-entropy method for cross-diffusion systems, Nonlinearity, 28 (2015), 1963-2001. doi: 10.1088/0951-7715/28/6/1963. Google Scholar

[29]

A. Jüngel and I. V. Stelzer, Entropy structure of a cross-diffusion tumor-growth model, Mathematical Models and Methods in Applied Sciences, 22 (2012), 1250009, 26pp. doi: 10.1142/S0218202512500091. Google Scholar

[30]

A. Jüngel and N. Zamponi, Qualitative behavior of solutions to cross-diffusion systems from population dynamics, Journal of Mathematical Analysis and Applications, 440 (2016), 794-809. doi: 10.1016/j.jmaa.2016.03.076. Google Scholar

[31]

A. Jüngel and N. Zamponi, Analysis of degenerate cross-diffusion population models with volume filling, Annales de l'Institut Henri Poincaré C, Analyse Non Linéaire, 34 (2017), 1-29. doi: 10.1016/j.anihpc.2015.08.003. Google Scholar

[32]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, Journal of Theoretical Biology, 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5. Google Scholar

[33]

H. Liu and H. Yu, The entropy satisfying discontinuous Galerkin method for Fokker-Planck equations, Journal of Scientific Computing, 62 (2015), 803-830. doi: 10.1007/s10915-014-9878-1. Google Scholar

[34]

H. Liu and Z. Wang, An entropy satisfying discontinuous Galerkin method for nonlinear Fokker–Planck equations, Journal of Scientific Computing, 68 (2016), 1217-1240. doi: 10.1007/s10915-016-0174-0. Google Scholar

[35]

H. Liu and Z. Wang, A free energy satisfying discontinuous Galerkin method for one-dimensional Poisson–Nernst–Planck systems, Journal of Computational Physics, 328 (2017), 413-437. doi: 10.1016/j.jcp.2016.10.008. Google Scholar

[36]

H. Liu and J. Yan, The direct discontinuous Galerkin (DDG) methods for diffusion problems, SIAM Journal on Numerical Analysis, 47 (2009), 675-698. doi: 10.1137/080720255. Google Scholar

[37]

A. A. H. Oulhaj, A finite volume scheme for a seawater intrusion model with cross-diffusion, in Finite Volumes for Complex Applications VIII – Methods and Theoretical Aspects (eds. C. Cancès and P. Omnes), Springer International Publishing, 199 (2017), 421–429. Google Scholar

[38]

W. H. Reed and T. Hill, Triangular mesh methods for the neutron transport equation, Technical report, Los Alamos Scientific Lab., N. Mex.(USA), 1973.Google Scholar

[39]

N. ShigesadaK. Kawasaki and E. Teramoto, Spatial segregation of interacting species, Journal of Theoretical Biology, 79 (1979), 83-99. doi: 10.1016/0022-5193(79)90258-3. Google Scholar

[40]

S. SrinivasanaJ. Poggiea and X. Zhang, A positivity-preserving high order discontinuous Galerkin scheme for convection-diffusion equations, Journal of Computational Physics, 366 (2018), 120-143. doi: 10.1016/j.jcp.2018.04.002. Google Scholar

[41]

Z. SunJ. A. Carrillo and C.-W. Shu, A discontinuous Galerkin method for nonlinear parabolic equations and gradient flow problems with interaction potentials, Journal of Computational Physics, 352 (2018), 76-104. doi: 10.1016/j.jcp.2017.09.050. Google Scholar

[42]

Z. Sun, J. A. Carrillo and C.-W. Shu, An entropy stable high-order discontinuous Galerkin method for cross-diffusion gradient flow systems, preprint, arXiv: 1810.03221.Google Scholar

[43]

X. Zhang, On positivity-preserving high order discontinuous Galerkin schemes for compressible Navier–Stokes equations, Journal of Computational Physics, 328 (2017), 301-343. doi: 10.1016/j.jcp.2016.10.002. Google Scholar

[44]

X. Zhang and C.-W. Shu, On maximum-principle-satisfying high order schemes for scalar conservation laws, Journal of Computational Physics, 229 (2010), 3091-3120. doi: 10.1016/j.jcp.2009.12.030. Google Scholar

Figure 1.  Numerical solutions to the tumor encapsulation problem in Example 4.2 with $ \beta = 0.0075 $ and $ \gamma = 10 $ at $ t = 0.02 $, $ t = 0.2 $, $ t = 1 $ and $ t = 2 $. We use piecewise cubic polynomials in the scheme. The mesh size is $ h = 0.04 $ in Figure 1a and Figure 1c, and is $ h = 0.02 $ in Figure 1b and Figure 1d. The corresponding entropy profiles are given in Figure 1e and Figure 1f. The reference solutions are given in black lines obtained by using the $ P^1 $ scheme on a mesh with $ h = 0.002 $
Figure 2.  Numerical solutions to the tumor encapsulation problem in Example 4.2 with $ \beta = 0.0075 $ and $ \gamma = 1000 $ at $ t = 0.02 $, $ t = 0.2 $, $ t = 1 $ and $ t = 2 $. The mesh size is $ h = 0.02 $ and the time step is $ \tau = 0.02h^2 $. The scaling parameter is set as $ 0.95\theta^ \varepsilon_{l, i} $ in the positivity-preserving procedure. Piecewise cubic polynomials are used in Figure 2a and Figure 2c, and piecewise quartic polynomials are used in Figure 2b and Figure 2d. The reference solution is given in black lines obtained by using the $ P^1 $ scheme on a mesh with $ h = 0.001 $
Figure 3.  Numerical solutions to the surfactant spreading problem in Example 4.3 at $ t = 1 $, $ t = 3 $ and $ t = 6 $. The mesh size is $ h = 0.05 $ and the time step is $ \tau = 0.02h^2 $. We apply piecewise cubic polynomials for producing Figure 3a and Figure 3c. Piecewise quartic polynomials are used in Figure 3b and Figure 3d. Positivity-preserving limiter is activated mainly near the leading front of $ \rho_2 $. The reference solutions are given in black lines obtained with $ P^1 $ scheme on a mesh with $ h = 0.0025 $. The profiles of discrete entropy are depicted in Figure 3e and Figure 3f
Figure 4.  Numerical solutions to the surfactant spreading problem (15) at $ t = 0.25 $. The solution is computed with piecewise cubic polynomials $ k = 3 $, mesh size $ h = 0.02 $ and time step $ \tau = 0.003h^2 $
Figure 5.  Numerical solutions to the seawater intrusion problem in Example 5.3 at $t = 0, 0.2, 0.79, 12$. Solutions are obtained with piecewise cubic polynomials on a uniform square mesh on $[0, 1]\times [0, 1]$, with $h^x = h^y =0.05$. The time step is set as $\tau = 0.002(h^x)^2$ in the simulation. $b$, $b+\rho_2$ and $b+\rho_1+\rho_2$ are depicted from bottom to top respectively
Table 1.  Accuracy test of the SKT population model in Example 4.1, with central flux for $\mathit{\boldsymbol{\hat \xi }} $ and Lax-Friedrichs flux for $ \widehat {F\mathit{\boldsymbol{u}}}$
$k$ $N$ $L^1$ error order $L^2$ error order $L^\infty$ error order
1 20 2.852E-02 - 1.009E-02 - 8.675E-03 -
40 9.370E-03 1.61 3.461E-03 1.54 2.918E-03 1.57
80 3.031E-03 1.63 1.149E-03 1.59 9.292E-04 1.65
160 9.527E-04 1.67 3.609E-04 1.67 2.765E-04 1.75
2 20 1.093E-03 - 4.283E-04 - 4.440E-04 -
40 1.022E-04 3.42 4.329E-05 3.31 4.291E-05 3.37
80 1.164E-05 3.13 5.223E-06 3.05 5.183E-06 3.05
160 1.414E-06 3.04 6.480E-07 3.01 6.428E-07 3.01
3 20 7.543E-05 - 2.840E-05 - 3.058E-05 -
40 8.282E-06 3.19 3.208E-06 3.15 3.901E-06 2.97
80 8.588E-07 3.27 3.501E-07 3.20 4.431E-07 3.14
160 8.748E-08 3.30 3.642E-08 3.27 4.812E-08 3.20
4 20 2.170E-06 - 9.649E-07 - 1.752E-06 -
40 3.209E-08 6.08 1.746E-08 5.79 3.606E-08 5.60
80 8.787E-10 5.19 5.031E-10 5.12 1.066E-09 5.08
160 2.620E-11 5.07 1.542E-11 5.03 3.288E-11 5.02
$k$ $N$ $L^1$ error order $L^2$ error order $L^\infty$ error order
1 20 2.852E-02 - 1.009E-02 - 8.675E-03 -
40 9.370E-03 1.61 3.461E-03 1.54 2.918E-03 1.57
80 3.031E-03 1.63 1.149E-03 1.59 9.292E-04 1.65
160 9.527E-04 1.67 3.609E-04 1.67 2.765E-04 1.75
2 20 1.093E-03 - 4.283E-04 - 4.440E-04 -
40 1.022E-04 3.42 4.329E-05 3.31 4.291E-05 3.37
80 1.164E-05 3.13 5.223E-06 3.05 5.183E-06 3.05
160 1.414E-06 3.04 6.480E-07 3.01 6.428E-07 3.01
3 20 7.543E-05 - 2.840E-05 - 3.058E-05 -
40 8.282E-06 3.19 3.208E-06 3.15 3.901E-06 2.97
80 8.588E-07 3.27 3.501E-07 3.20 4.431E-07 3.14
160 8.748E-08 3.30 3.642E-08 3.27 4.812E-08 3.20
4 20 2.170E-06 - 9.649E-07 - 1.752E-06 -
40 3.209E-08 6.08 1.746E-08 5.79 3.606E-08 5.60
80 8.787E-10 5.19 5.031E-10 5.12 1.066E-09 5.08
160 2.620E-11 5.07 1.542E-11 5.03 3.288E-11 5.02
Table 2.  Accuracy test of the SKT population model in Example 4.1, with alternating fluxes $ {\mathit{\boldsymbol{\hat \xi }}} = \boldsymbol{\xi}_h^- $ and $ \widehat {F\mathit{\boldsymbol{u}}} = (F_h \boldsymbol{u}_h)^+ $
$ k $ $ N $ $ L^1 $ error order $ L^2 $ error order $ L^\infty $ error order
1 20 8.476E-02 - 3.128E-02 - 2.359E-02 -
40 2.055E-02 2.04 7.346E-03 2.090 5.189E-03 2.18
80 5.088E-03 2.01 1.803E-03 2.027 1.235E-03 2.07
160 1.268E-03 2.00 4.486E-04 2.007 3.049E-04 2.02
2 20 1.313E-03 - 5.584E-04 - 6.613E-04 -
40 1.490E-04 3.14 6.531E-05 3.096 7.530E-05 3.14
80 1.815E-05 3.04 8.039E-06 3.022 9.240E-06 3.03
160 2.250E-06 3.01 1.001E-06 3.005 1.146E-06 3.01
3 20 4.008E-05 - 1.908E-05 - 3.948E-05 -
40 2.420E-06 4.05 1.160E-06 4.040 2.645E-06 3.90
80 1.503E-07 4.01 7.200E-08 4.010 1.682E-07 3.98
160 9.374E-09 4.00 4.493E-09 4.002 1.056E-08 3.99
4 20 1.603E-06 - 8.465E-07 - 1.774E-06 -
40 4.754E-08 5.08 2.615E-08 5.017 6.252E-08 4.83
80 1.469E-09 5.02 8.149E-10 5.004 2.028E-09 4.95
160 4.577E-11 5.00 2.544E-11 5.001 6.375E-11 4.99
$ k $ $ N $ $ L^1 $ error order $ L^2 $ error order $ L^\infty $ error order
1 20 8.476E-02 - 3.128E-02 - 2.359E-02 -
40 2.055E-02 2.04 7.346E-03 2.090 5.189E-03 2.18
80 5.088E-03 2.01 1.803E-03 2.027 1.235E-03 2.07
160 1.268E-03 2.00 4.486E-04 2.007 3.049E-04 2.02
2 20 1.313E-03 - 5.584E-04 - 6.613E-04 -
40 1.490E-04 3.14 6.531E-05 3.096 7.530E-05 3.14
80 1.815E-05 3.04 8.039E-06 3.022 9.240E-06 3.03
160 2.250E-06 3.01 1.001E-06 3.005 1.146E-06 3.01
3 20 4.008E-05 - 1.908E-05 - 3.948E-05 -
40 2.420E-06 4.05 1.160E-06 4.040 2.645E-06 3.90
80 1.503E-07 4.01 7.200E-08 4.010 1.682E-07 3.98
160 9.374E-09 4.00 4.493E-09 4.002 1.056E-08 3.99
4 20 1.603E-06 - 8.465E-07 - 1.774E-06 -
40 4.754E-08 5.08 2.615E-08 5.017 6.252E-08 4.83
80 1.469E-09 5.02 8.149E-10 5.004 2.028E-09 4.95
160 4.577E-11 5.00 2.544E-11 5.001 6.375E-11 4.99
Table 3.  Accuracy test of the SKT population model in Example 1 with $ k = 3 $, with central flux for $ {\mathit{\boldsymbol{\hat \xi }}} $ and the Lax-Friedrichs flux for $ \widehat {F\mathit{\boldsymbol{u}}} = \{F_h \boldsymbol{u}_h\} + \frac{\tilde{\alpha}}{2}[ \boldsymbol{\rho}_h] $. Here $ \tilde{\alpha} = 0, 10\alpha, 900\alpha $ respectively
$ \tilde{\alpha} $ $ N $ $ L^1 $ error order $ L^2 $ error order $ L^\infty $ error order
$ 0 $ 20 9.502E-05 - 3.419E-05 - 3.141E-05 -
40 1.196E-05 2.99 4.260E-06 3.00 4.207E-06 2.90
80 1.501E-06 2.99 5.331E-07 3.00 5.222E-07 3.01
160 1.878E-07 3.00 6.667E-08 3.00 6.516E-08 3.00
$ 10\alpha $ 20 3.941E-05 - 1.692E-05 - 2.287E-05 -
40 3.833E-06 3.36 1.644E-06 3.36 2.245E-06 3.35
80 3.351E-07 3.52 1.477E-07 3.48 2.136E-07 3.39
160 2.720E-08 3.62 1.238E-08 3.58 1.865E-08 3.52
$ 900\alpha $ 20 2.873E-06 - 1.096E-06 - 1.981E-06 -
40 1.558E-07 4.21 6.967E-08 3.98 1.213E-07 4.03
80 1.075E-08 3.86 4.892E-09 3.83 8.305E-09 3.87
160 7.043E-10 3.93 3.261E-10 3.91 5.546E-10 3.90
$ \tilde{\alpha} $ $ N $ $ L^1 $ error order $ L^2 $ error order $ L^\infty $ error order
$ 0 $ 20 9.502E-05 - 3.419E-05 - 3.141E-05 -
40 1.196E-05 2.99 4.260E-06 3.00 4.207E-06 2.90
80 1.501E-06 2.99 5.331E-07 3.00 5.222E-07 3.01
160 1.878E-07 3.00 6.667E-08 3.00 6.516E-08 3.00
$ 10\alpha $ 20 3.941E-05 - 1.692E-05 - 2.287E-05 -
40 3.833E-06 3.36 1.644E-06 3.36 2.245E-06 3.35
80 3.351E-07 3.52 1.477E-07 3.48 2.136E-07 3.39
160 2.720E-08 3.62 1.238E-08 3.58 1.865E-08 3.52
$ 900\alpha $ 20 2.873E-06 - 1.096E-06 - 1.981E-06 -
40 1.558E-07 4.21 6.967E-08 3.98 1.213E-07 4.03
80 1.075E-08 3.86 4.892E-09 3.83 8.305E-09 3.87
160 7.043E-10 3.93 3.261E-10 3.91 5.546E-10 3.90
Table 4.  Accuracy test of the cross-diffusion system in Example 5.1, with central flux for $ {\mathit{\boldsymbol{\hat \xi }}} $ and Lax-Friedrichs flux for $ \widehat {F\mathit{\boldsymbol{u}}} $
$ k $ $ N^x $ $ L^1 $ error order $ L^2 $ error order $ L^\infty $ error order
1 10 1.185E-01 - 5.349E-02 - 5.732E-02 -
20 4.656E-02 1.35 2.147E-02 1.32 2.265E-02 1.34
40 1.773E-02 1.39 8.266E-03 1.38 8.692E-03 1.38
80 6.218E-03 1.51 2.921E-03 1.50 3.084E-03 1.50
2 10 8.206E-03 - 4.362E-03 - 8.402E-03 -
20 9.124E-04 3.17 5.659E-04 2.95 9.942E-04 3.08
40 1.077E-04 3.08 7.186E-05 2.98 1.213E-04 3.04
80 1.315E-05 3.03 9.068E-06 2.99 1.515E-05 3.00
3 10 9.481E-04 - 5.256E-04 - 9.990E-04 -
20 1.061E-04 3.16 5.847E-05 3.17 1.112E-04 3.17
40 1.128E-05 3.23 6.169E-06 3.25 1.216E-05 3.19
80 1.119E-06 3.33 6.042E-07 3.35 1.232E-06 3.30
4 10 4.685E-05 - 2.649E-05 - 7.459E-05 -
20 1.212E-06 5.27 8.159E-07 5.02 2.837E-06 4.72
40 3.328E-08 5.19 2.273E-08 5.17 7.754E-08 5.19
$ k $ $ N^x $ $ L^1 $ error order $ L^2 $ error order $ L^\infty $ error order
1 10 1.185E-01 - 5.349E-02 - 5.732E-02 -
20 4.656E-02 1.35 2.147E-02 1.32 2.265E-02 1.34
40 1.773E-02 1.39 8.266E-03 1.38 8.692E-03 1.38
80 6.218E-03 1.51 2.921E-03 1.50 3.084E-03 1.50
2 10 8.206E-03 - 4.362E-03 - 8.402E-03 -
20 9.124E-04 3.17 5.659E-04 2.95 9.942E-04 3.08
40 1.077E-04 3.08 7.186E-05 2.98 1.213E-04 3.04
80 1.315E-05 3.03 9.068E-06 2.99 1.515E-05 3.00
3 10 9.481E-04 - 5.256E-04 - 9.990E-04 -
20 1.061E-04 3.16 5.847E-05 3.17 1.112E-04 3.17
40 1.128E-05 3.23 6.169E-06 3.25 1.216E-05 3.19
80 1.119E-06 3.33 6.042E-07 3.35 1.232E-06 3.30
4 10 4.685E-05 - 2.649E-05 - 7.459E-05 -
20 1.212E-06 5.27 8.159E-07 5.02 2.837E-06 4.72
40 3.328E-08 5.19 2.273E-08 5.17 7.754E-08 5.19
Table 5.  Accuracy test of the cross-diffusion system in Example 5.1 with alternating fluxes $\mathit{\boldsymbol{\hat \xi }} = \mathit{\boldsymbol{\xi }}_h^ - $ and $\widehat {F\mathit{\boldsymbol{u}}} = {\left( {{F_h}{\mathit{\boldsymbol{u}}_h}} \right)^ + }$
$k$ $N^x$ $L^1$ error order $L^2$ error order $L^\infty$ error order
1 10 3.848E-01 - 1.853E-01 - 2.415E-01 -
20 9.608E-02 2.00 4.397E-02 2.08 4.372E-02 2.47
40 2.376E-02 2.02 1.075E-02 2.03 1.014E-02 2.11
80 5.916E-03 2.01 2.667E-03 2.01 2.525E-03 2.01
2 10 1.510E-02 - 8.132E-03 - 2.121E-02 -
20 1.673E-03 3.17 9.563E-04 3.09 2.328E-03 3.19
40 1.923E-04 3.12 1.169E-04 3.03 2.662E-04 3.13
80 2.294E-05 3.07 1.453E-05 3.01 3.192E-05 3.06
3 10 8.253E-04 - 4.525E-04 - 1.449E-03 -
20 4.998E-05 4.05 2.922E-05 3.95 1.044E-04 3.80
40 3.075E-06 4.02 1.850E-06 3.98 6.675E-06 3.97
80 1.912E-07 4.01 1.161E-07 4.00 4.216E-07 3.99
4 10 5.174E-05 - 3.287E-05 - 1.448E-04 -
20 1.683E-06 4.94 1.127E-06 4.87 4.972E-06 4.86
40 5.276E-08 5.00 3.620E-08 4.96 1.485E-07 5.07
$k$ $N^x$ $L^1$ error order $L^2$ error order $L^\infty$ error order
1 10 3.848E-01 - 1.853E-01 - 2.415E-01 -
20 9.608E-02 2.00 4.397E-02 2.08 4.372E-02 2.47
40 2.376E-02 2.02 1.075E-02 2.03 1.014E-02 2.11
80 5.916E-03 2.01 2.667E-03 2.01 2.525E-03 2.01
2 10 1.510E-02 - 8.132E-03 - 2.121E-02 -
20 1.673E-03 3.17 9.563E-04 3.09 2.328E-03 3.19
40 1.923E-04 3.12 1.169E-04 3.03 2.662E-04 3.13
80 2.294E-05 3.07 1.453E-05 3.01 3.192E-05 3.06
3 10 8.253E-04 - 4.525E-04 - 1.449E-03 -
20 4.998E-05 4.05 2.922E-05 3.95 1.044E-04 3.80
40 3.075E-06 4.02 1.850E-06 3.98 6.675E-06 3.97
80 1.912E-07 4.01 1.161E-07 4.00 4.216E-07 3.99
4 10 5.174E-05 - 3.287E-05 - 1.448E-04 -
20 1.683E-06 4.94 1.127E-06 4.87 4.972E-06 4.86
40 5.276E-08 5.00 3.620E-08 4.96 1.485E-07 5.07
Table 6.  Accuracy test for Example 5.1, with central flux for ${\mathit{\boldsymbol{\hat \xi }}}$ and Lax-Friedrichs flux for $\widehat {F\mathit{\boldsymbol{u}}} = \{ {F_h}{\mathit{\boldsymbol{u}}_h}\} + \frac{{\tilde \alpha }}{2}\left[ {\mathit{\boldsymbol{\rho }}h} \right]$
$\tilde{\alpha}$ $N^x$ $L^1$ error order $L^2$ error order $L^\infty$ error order
0 10 1.082E-03 - 6.170E-04 - 1.139E-03 -
20 1.328E-04 3.07 7.682E-05 3.01 1.373E-04 3.05
40 1.651E-05 3.01 9.595E-06 3.00 1.722E-05 3.00
80 2.063E-06 3.00 1.199E-06 3.00 2.157E-06 3.00
$100\alpha$ 10 2.505E-04 - 1.266E-04 - 1.528E-04 -
20 1.234E-05 4.34 6.425E-06 4.30 1.234E-05 3.63
40 6.709E-07 4.20 3.609E-07 4.15 9.094E-07 3.76
80 3.923E-08 4.10 2.178E-08 4.05 6.480E-08 3.81
$\tilde{\alpha}$ $N^x$ $L^1$ error order $L^2$ error order $L^\infty$ error order
0 10 1.082E-03 - 6.170E-04 - 1.139E-03 -
20 1.328E-04 3.07 7.682E-05 3.01 1.373E-04 3.05
40 1.651E-05 3.01 9.595E-06 3.00 1.722E-05 3.00
80 2.063E-06 3.00 1.199E-06 3.00 2.157E-06 3.00
$100\alpha$ 10 2.505E-04 - 1.266E-04 - 1.528E-04 -
20 1.234E-05 4.34 6.425E-06 4.30 1.234E-05 3.63
40 6.709E-07 4.20 3.609E-07 4.15 9.094E-07 3.76
80 3.923E-08 4.10 2.178E-08 4.05 6.480E-08 3.81
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