## Abstract

This paper deals with nonlinear diffusion problems including the Stefan problem, the porous medium equation and cross-diffusion systems. A linear discrete-time scheme was proposed by Berger, Brezis and Rogers [*RAIRO Anal. Numér.* **13**
(1979) 297–312] for degenerate parabolic equations and was extended to cross-diffusion systems by Murakawa [*Math. Mod. Numer. Anal.* **45**
(2011) 1141–1161]. There is a constant stability parameter \(\mu \) in the linear scheme. In this paper, we propose a linear discrete-time scheme replacing the constant \(\mu \) with given functions depending on time, space and species. After discretizing the scheme in space, we obtain an easy-to-implement numerical method for the nonlinear diffusion problems. Convergence rates of the proposed discrete-time scheme with respect to the time increment are analyzed theoretically. These rates are the same as in the case where \(\mu \) is constant. However, actual errors in numerical computation become significantly smaller if varying \(\mu \) is employed. Our scheme has many advantages even though it is very easy-to-implement, e.g., the ensuing linear algebraic systems are symmetric, it requires low computational cost, the accuracy is comparable to that of the well-studied nonlinear schemes, the computation is much faster than the nonlinear schemes to obtain the same level of accuracy.

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## Change history

### 16 January 2018

The author would like to correct the figures and citations in the publication of the original article.

## References

Andreianov, B., Bendahmane, M., Ruiz-Baier, R.: Analysis of a finite volume method for a cross-diffusion model in population dynamics. Math. Models Methods Appl. Sci.

**21**, 307–344 (2011)Barenblatt, G.I.: On some unsteady motion of a liquid or a gas in a porous medium. Prikl. Math. Meh.

**16**, 67–78 (1952)Barrett, J.W., Blowey, J.F.: Finite element approximation of a nonlinear cross-diffusion population model. Numer. Math.

**98**, 195–221 (2004)Beckett, G., Mackenzie, J.A., Robertson, M.L.: A moving mesh finite element method for the solution of two-dimensional Stefan problems. J. Comput. Phys.

**168**, 500–518 (2001)Berger, A.E., Brezis, H., J.C.W., Rogers: A numerical method for solving the problem \(u_t-\Delta f(u)=0\). R.A.I.R.O. Anal. Numér

**13**, 297–312 (1979)Chen, L., Jüngel, A.: Analysis of a multidimensional parabolic population model with strong cross-diffusion. SIAM J. Math. Anal.

**36**, 301–322 (2006)Crank, J.: Free and Moving Boundary Problems. Clarendon Press, Oxford (1984)

Dreher, M.: Analysis of a population model with strong cross-diffusion in unbounded domains. Proc. R. Soc. Edinb. Sect. A

**138**, 769–786 (2008)Elliott, C.M.: Error analysis of the enthalpy method for the Stefan problem. IMA J. Numer. Anal.

**7**, 61–71 (1987)Furzeland, R.M.: A comparative study of numerical methods for moving boundary problems. J. Inst. Maths Appl.

**26**, 411–429 (1980)Galiano, G., Garzón, M.L., Jüngel, A.: Semi-discretization in time and numerical convergence of solutions of a nonlinear cross-diffusion population model. Numer. Math.

**93**, 655–673 (2003)Jäger, W., Kačur, J.: Solution of porous medium type systems by linear approximation schemes. Numer. Math.

**60**, 407–427 (1991)Jerome, J.W., Rose, M.E.: Error estimates for the multidimensional two-phase Stefan problem. Math. Comput.

**39**(160), 377–414 (1982)Mackenzie, J.A., Robertson, M.L.: The numerical solution of one-dimensional phase change problems using an adaptive moving mesh method. J. Comput. Phys.

**161**, 537–557 (2000)Magenes, E., Nochetto, R.H., Verdi, C.: Energy error estimates for a linear scheme to approximate nonlinear parabolic problems. Math. Model. Numer. Anal.

**21**, 655–678 (1987)Magenes, E., Verdi, C., Visintin, A.: Theoretical and numerical results on the two-phase Stefan problem. SIAM J. Numer. Anal.

**26**, 1425–1438 (1989)Murakawa, H.: A linear scheme to approximate nonlinear cross-diffusion systems. Math. Model. Numer. Anal.

**45**, 1141–1161 (2011)Murakawa, H.: Numerical solution of nonlinear cross-diffusion systems by a linear scheme. In: Kawashima, S., Ei, S., Kimura, M., Mizumachi, T. (eds.) Proceedings for the 4th MSJ-SI conference on nonlinear dynamics in partial differential equations, Adv. Stud. Pure Math., vol. 64, pp. 243–251. (2015)

Murakawa, H.: Error estimates for discrete-time approximations of nonlinear cross-diffusion systems. SIAM J. Numer. Anal.

**52**(2), 955–974 (2014)Murakawa, H.: A linear finite volume method for nonlinear cross-diffusion systems. Numer. Math.

**136**(1), 1–26 (2017)Nochetto, R.H.: Error estimates for two-phase Stefan problems in several space variables, I: linear boundary conditions. Calcolo

**22**, 457–499 (1985)Nochetto, R.H., Paolini, M., Verdi, C.: An adaptive finite element method for two-phase Stefan problems in two space dimensions. Part I: stability and error estimates. Math. Comput.

**57**(195), 73–108 (1991)Nochetto, R.H., Paolini, M., Verdi, C.: A fully discrete adaptive nonlinear Chernoff formula. SIAM J. Numer. Anal.

**30**, 991–1014 (1993)Nochetto, R.H., Verdi, C.: An efficient linear scheme to approximate parabolic free boundary problems: error estimates and implementation. Math. Comput.

**51**, 27–53 (1988)Pop, I.S., Yong, W.A.: A numerical approach to degenerate parabolic equations. Numer. Math.

**92**, 357–381 (2002)Shigesada, N., Kawasaki, K., Teramoto, E.: Spatial segregation of interacting species. J. Theor. Biol.

**79**, 83–99 (1979)Verdi, C.: Optimal error estimates for an approximation of degenerate parabolic problems. Numer. Funct. Anal. Optim.

**9**, 657–670 (1987)Verdi, C.: Numerical aspects of parabolic free boundary and hysteresis problems, Lecture Notes in Mathematics, vol. 1584, pp. 213–284. Springer, Berlin (1994)

Zamponi, N., Jüngel, A.: Analysis of degenerate cross-diffusion population models with volume filling. Ann. Inst. H. Poincaré Anal. Non Linéaire

**34**, 1–29 (2017)

## Acknowledgements

This work was supported by JSPS KAKENHI Grant nos. 26287025, 15H03635 and 17K05368, and JST CREST Grant No. JPMJCR14D3.

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The original version of this article was revised: The figures, captions and the text citations were revised.

A correction to this article is available online at https://doi.org/10.1007/s13160-017-0283-7.

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Murakawa, H. An efficient linear scheme to approximate nonlinear diffusion problems.
*Japan J. Indust. Appl. Math.* **35**, 71–101 (2018). https://doi.org/10.1007/s13160-017-0279-3

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DOI: https://doi.org/10.1007/s13160-017-0279-3

### Keywords

- Stefan problem
- Porous medium equation
- Cross-diffusion system
- Linear scheme
- Error estimate
- Numerical method