Modified Rosenbrock methods with approximate Jacobian matrices

Exponential Rosenbrock-Type Methods

We introduce a new class of exponential integrators for the numerical integration of large-scale systems of stiff differential equations. These so-called Rosenbrock-type methods linearize the flow in each time step and make use of the matrix exponential and related functions of the Jacobian. In contrast to standard integrators, the methods are fully explicit and do not require the numerical sol...

Two modified Jacobi methods for M-matrices

Parallel exponential Rosenbrock methods

Exponential Rosenbrock integrators have been shown to be very efficient in solving large stiff differential systems of ODEs. So far, such exponential methods have been derived up to order 5. In this talk we give a convergence result for methods of order up to 6 and construct new integrators of orders 4, 5, and 6. In contrast to the existing schemes of orders 4 and 5, the new schemes, which are ...

two modified jacobi methods for m-matrices

Nilpotent Symmetric Jacobian Matrices and the Jacobian Conjecture Ii

It is shown that the Jacobian Conjecture holds for all polynomial maps F : k → k of the form F = x + H , such that JH is nilpotent and symmetric, when n ≤ 4. If H is also homogeneous a similar result is proved for all n ≤ 5. Introduction Let F := (F1, . . . , Fn) : C → C be a polynomial map i.e. each Fi is a polynomial in n variables over C. Denote by JF := (i ∂xj )1≤i,j≤n, the Jacobian matrix ...