Second-order Boltzmann schemes for compressible Euler equations in one and two space dimensions
SIAM Journal on Numerical Analysis
Numerical Navier-Stokes solutions from gas kinetic theory
Journal of Computational Physics
An analysis of numerical errors in large-eddy simulations of turbulence
Journal of Computational Physics
A space-time formulation for multiscale phenomena
Journal of Computational and Applied Mathematics - Special issue on TICAM symposium
Optimal prediction for Hamiltonian partial differential equations
Journal of Computational Physics
Coupling of atomistic and continuum simulations using a bridging scale decomposition
Journal of Computational Physics
Finite difference heterogeneous multi-scale method for homogenization problems
Journal of Computational Physics
Heterogeneous multiscale method for the modeling of complex fluids and micro-fluidics
Journal of Computational Physics
Optimal spatiotemporal reduced order modeling, Part II: application to a nonlinear beam
Computational Mechanics
Optimal spatiotemporal reduced order modeling, Part II: application to a nonlinear beam
Computational Mechanics
Characterization of subgrid-scale dynamics for a nonlinear beam
Computers and Structures
Optimal spatiotemporal reduced order modeling of the viscous Burgers equation
Finite Elements in Analysis and Design
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An optimal spatiotemporal reduced order modeling framework is proposed for nonlinear dynamical systems in continuum mechanics. In this paper, Part I, the governing equations for a general system are modified for an under-resolved simulation in space and time with an arbitrary discretization scheme. Basic filtering concepts are used to demonstrate the manner in which subgrid-scale dynamics arise with a coarse computational grid. Models are then developed to account for the underlying spatiotemporal structure via inclusion of statistical information into the governing equations on a multi-point stencil. These subgrid-scale models are designed to provide closure by accounting for the interactions between spatiotemporal microscales and macroscales as the system evolves. Predictions for the modified system are based upon principles of mean-square error minimization, conditional expectations and stochastic estimation, thus rendering the optimal solution with respect to the chosen resolution. Practical methods are suggested for model construction, appraisal, error measure and implementation. The companion paper, Part II, is devoted to demonstrating the methodology through a computational study of a nonlinear beam.