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A strongly A-stable time integration method for solving the nonlinear reaction-diffusion equation. (English) Zbl 1352.65204
Summary: The semidiscrete ordinary differential equation (ODE) system resulting from compact higher-order finite difference spatial discretization of a nonlinear parabolic partial differential equation, for instance, the reaction-diffusion equation, is highly stiff. Therefore numerical time integration methods with stiff stability such as implicit Runge-Kutta methods and implicit multistep methods are required to solve the large-scale stiff ODE system. However those methods are computationally expensive, especially for nonlinear cases. Rosenbrock method is efficient since it is iteration-free; however it suffers from order reduction when it is used for nonlinear parabolic partial differential equation. In this work we construct a new fourth-order Rosenbrock method to solve the nonlinear parabolic partial differential equation supplemented with Dirichlet or Neumann boundary condition. We successfully resolved the phenomena of order reduction, so the new method is fourth-order in time when it is used for nonlinear parabolic partial differential equations. Moreover, it has been shown that the Rosenbrock method is strongly A-stable hence suitable for the stiff ODE system obtained from compact finite difference discretization of the nonlinear parabolic partial differential equation. Several numerical experiments have been conducted to demonstrate the efficiency, stability, and accuracy of the new method.

##### MSC:
 65L20 Stability and convergence of numerical methods for ordinary differential equations 35K57 Reaction-diffusion equations
RODAS; ROS3P
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##### References:
 [1] Adam, Y., Highly accurate compact implicit methods and boundary conditions, Journal of Computational Physics, 24, 1, 10-22, (1977) · Zbl 0357.65074 [2] Mitchell, A. R.; Griffths, D. F., The Finite Difference Method in Partial Differential Equations, (1980), New York, NY, USA: John Wiley & Sons, New York, NY, USA [3] Ramos, J. I., Linearization methods for reaction-diffusion equations: multidimensional problems, Applied Mathematics and Computation, 88, 2-3, 225-254, (1997) · Zbl 0904.65089 [4] Ramos, J. I., Implicit, compact, linearized $$\theta$$-methods with factorization for multidimensional reaction-diffusion equations, Applied Mathematics and Computation, 94, 1, 17-43, (1998) · Zbl 0943.65098 [5] Chu, P. C.; Fan, C., A three-point combined compact difference scheme, Journal of Computational Physics, 140, 2, 370-399, (1998) · Zbl 0923.65071 [6] Luan, V.; Ostermann, A., Exponential Rosenbrock methods of order five—construction, analysis and numerical comparisons, Journal of Computational and Applied Mathematics, 255, 1, 417-431, (2014) · Zbl 1291.65201 [7] Luan, V. T.; Ostermann, A., Explicit exponential Runge-Kutta methods of high order for parabolic problems, Journal of Computational and Applied Mathematics, 256, 168-179, (2014) · Zbl 1314.65103 [8] Kassam, A.-K.; Trefethen, L. N., Fourth-order time-stepping for stiff PDEs, SIAM Journal on Scientific Computing, 26, 4, 1214-1233, (2005) · Zbl 1077.65105 [9] Rosenbrock, H., Some general implicit processes for the numerical solution of differential equations, The Computer Journal, 5, 329-330, (1963) · Zbl 0112.07805 [10] Haines, C. F., Implicit integration processes with error estimate for the numerical solution of differential equations, The Computer Journal, 12, 2, 183-188, (1969) · Zbl 0185.41203 [11] Liao, W.; Yan, Y., Singly diagonally implicit Runge-Kutta method for time-dependent reaction-diffusion equation, Numerical Methods for Partial Differential Equations, 27, 6, 1423-1441, (2011) · Zbl 1243.65112 [12] Dahlquist, G. G., A special stability problem for linear multistep methods, BIT Numerical Mathematics, 3, 27-43, (1963) · Zbl 0123.11703 [13] Gourlay, A. R., A note on trapezoidal methods for the solution of initial value problems, Mathematics of Computation, 24, 629-633, (1970) · Zbl 0233.65040 [14] Hairer, E.; Wanner, G., Solving Ordinary Differential Equations, II, Stiff and Algebraic Problems, (1996), Berlin , Germany: Springer, Berlin , Germany · Zbl 1192.65097 [15] Alexander, R., Diagonally implicit Runge-Kutta methods for stiff O.D.E.’s, SIAM Journal on Numerical Analysis, 14, 6, 1006-1021, (1977) · Zbl 0374.65038 [16] Burrage, K.; Butcher, J. C.; Chipman, F. H., An implementation of singly-implicit Runge-Kutta methods, BIT Numerical Mathematics, 20, 3, 326-340, (1980) · Zbl 0456.65040 [17] Claus, H., Singly-implicit Runge-Kutta methods for retarded and ordinary differential equations, Computing, 43, 3, 209-222, (1990) · Zbl 0695.65047 [18] Liniger, W.; Willoughby, R. A., Efficient integration methods for stiff systems of ordinary differential equations, SIAM Journal on Numerical Analysis, 7, 47-66, (1970) · Zbl 0187.11003 [19] Verwer, J. G.; Spee, E. J.; Blom, J. G.; Hundsdorfer, W., A second-order Rosenbrock method applied to photochemical dispersion problems, SIAM Journal on Scientific Computing, 20, 4, 1456-1480, (1999) · Zbl 0922.65031 [20] Lang, J.; Verwer, J., ROS3P—an accurate third-order Rosenbrock solver designed for parabolic problems, BIT Numerical Mathematics, 41, 4, 731-738, (2001) · Zbl 0996.65099 [21] Hundsdorfer, W. H., Stability and B-convergence of linearly implicit Runge-Kutta methods, Numerische Mathematik, 50, 1, 83-95, (1986) · Zbl 0611.65045 [22] Lubich, C.; Ostermann, A., Linearly implicit time discretization of non-linear parabolic equations, IMA Journal of Numerical Analysis, 15, 4, 555-583, (1995) · Zbl 0834.65092 [23] Kaps, P.; Rentrop, P., Generalized Runge-Kutta methods of order four with stepsize control for stiff ordinary differential equations, Numerische Mathematik, 33, 1, 55-68, (1979) · Zbl 0436.65047 [24] Shampine, L. F., Implementation of Rosenbrock methods, ACM Transactions on Mathematical Software, 8, 2, 93-113, (1982) · Zbl 0483.65041 [25] van Veldhuizen, M., $$D$$-stability and Kaps-Rentrop-methods, Computing. Archives for Scientific Computing, 32, 3, 229-237, (1984) · Zbl 0528.65041 [26] Bui, T. D., On an $$L$$-stable method for stiff differential equations, Information Processing Letters, 6, 5, 158-161, (1977) · Zbl 0412.65037
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