Stable time integration schemes with high order of accuracy

Numerical formulations of coupled problems are typically prone to dynamic instabilities arising from the nonlinear interactions between field variables. It should be obvious that time integration schemes that are even unconditionally stable for linear problems, lose this property when applied to initial and boundary value problems on nonlinear, multifield pdes. Numerical efficiency is further served if, in addition to unconditional stability, the schemes can be designed to be of high order in time. One approach to developing such methods relies on an redefinition of the time-discrete form of constitutive relations for quantities derived from the primal field, and in terms of which the balance equations are posed. These include stresses obtained from gradients of the displacement field, and chemical potentials obtained from the composition. Such unconditionally stable, second-order time integration schemes are being developed in our group for a range of nonlinear, multiphysics problems. Of particular interest are those involving high-order spatial derivatives, such as variants of phase field models and gradient elasticity.


Publications:

K. Sagiyama, S. Rudraraju and K. Garikipati, ``Unconditionally stable, second-order accurate schemes for solid state phase transformations driven by mechano-chemical spinodal decomposition'',(under review). [arXiv]

The collage below shows microstructures that form in a material undergoing mechano-chemical spinodal decomposition, the unconditionally non-growing free energy, and evidence of the second-order convergence with time.

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