Several Structure-Preserving Numerical Methods For Solving Some Classes Of Stochastic Differential Equations On Manifolds

The theory of stochastic differential equation is not only the supplement to the theory of deterministic differential equation,but also the performance of a deeper understanding of the nature of the real world.However,the complexity of the stochastic differential equation makes it always dependent on numerical methods to solve it.It is a broad consensus in the research area of numerical computation that the natural property of the original problem should be preserved after discretization.From this point of view,the geometric numerical methods for stochastic differential equations on manifold(with conserved quantity or on Lie group)are constructed.Main contributions of this dissertation are as follows:The projection method is one of the most classic construction ideas of the conservedquantity-preserving numerical method.The advantage of the stochastic standard projection method is that it can be directly applied to solve stochastic differential equations with multiple conserved quantities,but the disadvantage is that it needs to calculate the nonlinear equations,which increases the amount of calculation.To this end,with the help of local coordinate parameterization,the stochastic differential equation is transformed into the stochastic differential equation under the local coordinate and solved by numerical method,and then the obtained numerical solution is mapped back to the manifold where the original equation is located by the local coordinate transformation to obtain the multiple-conserved-quantity-preserving method.With the help of the concept of discrete tangent space,the single-step increment of the numerical method for stochastic differential equations is projected onto the discrete tangent space determined by the discrete gradient of the original equation’s conserved quantity,thereby the multiple-conservedquantity-preserving method is constructed,and the projection operator is computed by QR composition.Another disadvantage of the stochastic standard projection method is that it may destroy the geometric properties such as the symmetry of the underlying method.The stochastic symmetric projection method which maintains the symmetry of the underlying method is constructed.The definition of ρ-reversibility of the stochastic differential equation is given,the relationship between ρ-reversibility and symmetry is illustrated,and the sufficient conditions for ρ-reversibility of the stochastic symmetric projection method are proved.Besides,by giving a single conserved quantity form corresponding to the multiple conserved quantities of the original equation,the nearly multiple-conservedquantity-preserving method is constructed.The stochastic discrete gradient method is another class of conserved-quantitypreserving method for stochastic differential equations.The advantage of this method is that it can make the numerical method have good characteristics such as symmetry while maintaining the conserved quantity of the original equation,since the choice of oblique gradient form and discrete gradient in the method is arbitrary.But the disadvantage of the stochastic discrete gradient method is that it can only preserve a single conserved quantity of the original equation.The stochastic discrete gradient method is generalized to preserve multiple conserved quantities of the original equation.Magnus method is a classic numerical method for preserving Lie group structure.The advantage of the Magnus method is even if appropriately truncating the expansions,the result still maintain the properties of the original equations.The stochastic Magnus expansion of stochastic differential equations is given by using the colored root tree theory.By truncating the stochastic Magnus expansion,the stochastic Magnus method for solving linear stochastic differential equations on Lie groups is constructed.With the help of Lie derivative operator,the construction idea of stochastic Magnus method is extended to nonlinear stochastic differential equations.For the commutative case,the arbitrarily high order numerical method is constructed by using the stochastic Magnus expansion.The construction idea of Lie group method is extended to stochastic differential equations on general manifolds.With the help of concepts such as φ-relatedness and Lie algebra action,the stochastic Munthe-Kaas method for solving stochastic differential equations is constructed.The definition of(graded)free Lie algebra is given by using the Hall set,and by calculating in the free Lie algebra,the number of Lie brackets needed to be calculated during the implementation of the method is reduced,thereby improving the computational efficiency of the numerical method.The mean-square convergence and the property of preserving geometric structure of the proposed numerical methods are analyzed theoretically.Numerical examples are used to verify the validity of the constructed numerical method and the correctness of the theory.