Efficient stability-preserving numerical methods for nonlinear coercive problems in vector space. (English) Zbl 1529.65044
Summary: Strong stability (or monotonicity)-preserving time discretization schemes preserve the stability properties of the exact solution and have proved very useful in scientific and engineering computation, especially in solving hyperbolic partial differential equations. The main aim of this work is to further extend this to exponential stability-preserving numerical methods for a general coercive system whose solution is exponentially growing or decaying and the rate of growth or decay can be quantified by a \((\omega,\tau^*)\) function in general vector space with a convex functional. Under the same stepsize condition as for strong stability, sharper exponential stability results are derived for explicit and diagonally implicit Runge-Kutta methods and variable coefficients linear multistep methods for nonlinear problems. The new developments in this paper also include their applications to various linear and nonlinear evolution problems.
MSC:
| 65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
| 65L03 | Numerical methods for functional-differential equations |
| 65L70 | Error bounds for numerical methods for ordinary differential equations |
| 65L20 | Stability and convergence of numerical methods for ordinary differential equations |
| 65N06 | Finite difference methods for boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65M50 | Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
Keywords:
nonlinear evolution equations; coercive problems; Runge-Kutta methods; linear multistep methods; stability; strong stability; exponential stability; monotonicity; contractivity; vector space; convex functional; time discretizationSoftware:
RODASReferences:
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