Accelerated solutions of convection‐dominated partial differential equations using implicit feature tracking and empirical quadrature

Summary This work introduces an empirical quadrature‐based hyperreduction procedure and greedy training algorithm to effectively reduce the computational cost of solving convection‐dominated problems with limited training. The proposed approach circumvents the slowly decaying n$$ n $$‐width limitation of linear model reduction techniques applied to convection‐dominated problems by using a nonlinear approximation manifold systematically defined by composing a low‐dimensional affine space with bijections of the underlying domain. The reduced‐order model is defined as the solution of a residual minimization problem over the nonlinear manifold. An online‐efficient method is obtained by using empirical quadrature to approximate the optimality system such that it can be solved with mesh‐independent operations. The proposed reduced‐order model is trained using a greedy procedure to systematically sample the parameter domain. The effectiveness of the proposed approach is demonstrated on two shock‐dominated computational fluid dynamics benchmarks. This work introduces an empirical quadrature‐based hyperreduction procedure and greedy training algorithm to effectively reduce the computational cost of solving convection‐dominated problems with limited training. The reduced‐order model is defined as the solution of a residual minimization problem over a nonlinear trial manifold. An online‐efficient method is obtained by using empirical quadrature to approximate the optimality system such that it can be solved with mesh‐independent operations.