A Reduced-order Backward Euler Finite Element Scheme For Two Kinds Of Wave Equations Based On POD Method

Many problems encountered in scientific research of different disciplines such as physics,engineering,medicine and economics are represented by partial differential e-quations.To obtain useful data and predictions,they shall be solved,whereas solutions to most partial differential equations can be hardly presented in practical analytical forms.Thus,numerical solutions are essential for solving these equations and some-what make up the above deficiency.However,there are also some limitations upon numerical methods.In solving complex partial differential equations,an extremely high degree of freedom is always required however good the discreet schemes are.As a consequence,considerable costs must be paid for the memory and the calculations.Therefore,it is quite necessary to study how to reduce computation,control trunca-tion errors,save computation time and lower memory standards on the premise that numerical solutions to the equations are precise enough.Dimension reduction is one of effective ways for solving this problem,while proper orthogonal decomposition,as a relatively well-known method for reducing dimensions,has been successfully used for dimension reduction in complex system models.Its essence is low-dimensional approx-imate descriptions of physical processes,where the optimal values are approximate to the known data,in order to really simplify computations,save computation time and reduce memory.This paper focuses on studying two aspects as follows:First of all,it mainly applies proper orthogonal decomposition method in gener-al Euler-FE schemes for BBM-Burgers equations to reduce their computation load,and extracts sets of image sequences from the finite element solutions.Subsequent-ly,the finite element space of the finite element schemes is replaced by POD-based space.Higher-dimensional Euler-FE schemes are simplified and converted into lower-dimensional POD-based backward ones with enough high precisions.Meanwhile,er-rors in reduced-order Euler-FE are estimated.Furthermore,this paper illustrates how to develop a general Euler-FE scheme for Rosenau-RLW solutions based on proper orthogonal decomposition,simplifies it into a relatively precise POD-based backward Euler-FE scheme,and estimates errors in the simplified scheme.The POD-based backward Euler-FE schemes are found to be more effective for the general schemes.