Galerkin/Runge-Kutta discretizations for semilinear parabolic equations

A new class of fully discrete Galerkin/Runge–Kutta methods is constructed and analyzed for semilinear parabolic initial boundary value problems. Unlike any classical counterpart, this class offers arbitrarily high-order convergence without suffering from what has been called order reduction. In support of this claim, error estimates are proved, and computational results are presented. Furthermore, it is noted that special Runge–Kutta methods allow computations to be performed in parallel so that the final execution time can be reduced to that of a low-order method.

[1]  Vidar Thomée,et al.  Single step Galerkin approximations for parabolic problems , 1977 .

[2]  Vidar Thomée,et al.  Some Convergence Estimates for Semidiscrete Galerkin Type Approximations for Parabolic Equations , 1977 .

[3]  Ohannes A. Karakashian,et al.  On multistep-Galerkin discretizations of semilinear hyperbolic and parabolic equations , 1980 .

[4]  J. Verwer,et al.  Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations , 1984 .

[5]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[6]  A. H. Schatz,et al.  On the quasi-optimality in _{∞} of the ¹-projection into finite element spaces , 1982 .

[7]  Mary Fanett.  A PRIORI L2 ERROR ESTIMATES FOR GALERKIN APPROXIMATIONS TO PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS , 1973 .

[8]  Ohannes A. Karakashian,et al.  On Runge-Kutta methods for parabolic problems with time-dependent coefficients , 1986 .

[9]  M. Wheeler.  A Priori L_2 Error Estimates for Galerkin Approximations to Parabolic Partial Differential Equations , 1973 .

[10]  J. Butcher.  Implicit Runge-Kutta processes , 1964 .

[11]  Galerkin's method for some highly nonlinear problems , 1977 .

[12]  S. Keeling.  On Lipschitz continuity of nonlinear differential operators , 1990 .

[13]  Rolf Rannacher,et al.  Some Optimal Error Estimates for Piecewise Linear Finite Element Approximations , 1982 .

[14]  Ohannes A. Karakashian,et al.  On a higher order accurate fully discrete Galerkin approximation to the Navier-Stokes equations , 1982 .

[15]  Ohannes A. Karakashian,et al.  On some high-order accurate fully discrete Galerkin methods for the Korteweg-de Vries equation , 1985 .

[16]  A. H. Schatz,et al.  On the Quasi-Optimality in $L_\infty$ of the $\overset{\circ}{H}^1$-Projection into Finite Element Spaces* , 1982 .