Nonlinear energy-preserving model reduction with lifting transformations that quadratize the energy
This work addresses a domain-specific problem for researchers in computational physics and engineering dealing with high-dimensional conservative PDEs, offering incremental improvements in offline computational gains.
The paper tackles the computational bottleneck in model reduction for conservative PDEs with non-polynomial nonlinearities by introducing a lifting transformation method that quadratizes the energy, resulting in quadratic reduced-order models that conserve energy exactly and show competitive accuracy and efficiency compared to state-of-the-art methods.
Existing model reduction techniques for high-dimensional models of conservative partial differential equations (PDEs) encounter computational bottlenecks when dealing with systems featuring non-polynomial nonlinearities. This work presents a nonlinear model reduction method that employs lifting variable transformations to derive structure-preserving quadratic reduced-order models for conservative PDEs with general nonlinearities. We present an energy-quadratization strategy that defines the auxiliary variable in terms of the nonlinear term in the energy expression to derive an equivalent quadratic lifted system with quadratic system energy. The proposed strategy combined with proper orthogonal decomposition model reduction yields quadratic reduced-order models that conserve the quadratized lifted energy exactly in high dimensions. We demonstrate the proposed model reduction approach on four nonlinear conservative PDEs: the one-dimensional wave equation with exponential nonlinearity, the two-dimensional sine-Gordon equation, the two-dimensional Klein-Gordon equation with parametric dependence, and the two-dimensional Klein-Gordon-Zakharov equations. The numerical results show that the proposed lifting approach is competitive with the state-of-the-art structure-preserving hyper-reduction method in terms of both accuracy and computational efficiency in the online stage while providing significant computational gains in the offline stage.