A Reduced-order Extrapolated Model Based On Proper Orthogonal Decomposition Method For Hyperbolic Equation

Proper orthogonal decomposition(POD),as an efficient method of dimensionality reduction,has been widely used in practical engineering problems because of its ability to rapidly solve the numerical solutions of partial differential equations in combination with some numerical methods.This study is mainly concerned with the reduced-order extrapolating method about the unknown solution coefficient vectors in the mixed finite element(MFE)method for the linear fourth-order hyperbolic equation and a class of nonlinear damped Boussinesq equation.For this purpose,the matrix-model of the MFE method and the stability and error estimates about the MFE matrix model solutions are first derived.Secondly,a reduced-order extrapolating MFE(ROEMFE)model in matrix-form is established with two POD basis vectors are generated by the first several coefficient vectors of MFE solution,and the existence,stability,and error estimates of the ROEMFE solutions are proved by matrix theory,which makes the theoretical analysis very simple.In particular,in the study of the nonlinear damped Boussinesq equation,the properties of the original variable and the intermediate variable are derived simultaneously,making the theory well developed.Finally,numerical experiments show that the error order of ROEMFE method and MFE method is basically the same,but ROEMFE method can greatly reduce the degree of freedom,so that the ROEMFE method does have a great advantage in CPU running time,which proves that ROEMFE method is very effective in solving time-dependent equations similar to the fourth order hyperbolic equations.