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Semi-implicit discretization of abstract evolution equations. (English) Zbl 0807.65060
The authors consider a semi-implicit discretization of the following abstract Cauchy problem in a Banach space \(X: u'= f(t,u,u')\), \(t\in (0,T]\), \(u(0)= u_ 0\), where \(f(t,\cdot,v)\) is one-sided Lipschitz, \(R(I- hf(t,\cdot,v))= X\) for \(h>0\) sufficiently small and \(f(t,u,\cdot)\) is Lipschitz continuous. Using the implicit Euler method, the authors deduce \(u_{n+1}= u_ n+ hf(t_{n+1},u_{n+1},u_ n)\), with \(t_ n= nh\), \(h= T/N\),\(u_ n= u(t_ n)\) and show that the latter is stable and consistent, and hence convergent.
The proof of the above-mentioned results will be published in another paper. Moreover, they apply the algorithm to semilinear parabolic (integro-)differential systems in both reflexive and non-reflexive spaces but present no numerical results.
65J15 Numerical solutions to equations with nonlinear operators
65L05 Numerical methods for initial value problems involving ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
65R20 Numerical methods for integral equations
45N05 Abstract integral equations, integral equations in abstract spaces
34G20 Nonlinear differential equations in abstract spaces
35K57 Reaction-diffusion equations
45J05 Integro-ordinary differential equations
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