Lamoureux, Michael P.Vestrum, Robert J.2021-05-172021-05-172021-05-14Vestrum, R. J. (2021). Local Factorization of Multidimensional Differential Operators to Optimize Implicit Solution Methods (Master's thesis, University of Calgary, Calgary, Canada). Retrieved from https://prism.ucalgary.ca.http://hdl.handle.net/1880/113427Solving multidimensional differential operators using implicit finite-difference methods involves a computationally intensive step of calculating the solution to a system that involves both the current and future state of the system. If a linear operator can be expressed as an affine combination of sufficiently many orthogonal finite-difference approximations, then it is possible to factor the operator as product of an upper triangular matrix and that matrix's adjoint. Using the factor calculated in this Affine Local Grid Factorization (ALG-F), it is possible to solve the system by simple back substitution followed by forward substitution resulting in an implicit scheme that is O(N) rather than O(N^2) typical in implicit finite-difference schemes. Included are the equations to calculate the ALG Factorization for the wave equation in two and three dimensions to demonstrate the method provides robust and accurate results.engUniversity of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission.Hyperbolic PDEPartial Differential EquationImplicit Finite Difference MethodLocal GridBack-SubstitutionFinite-Difference MethodFactorizationLaplace OperatorEducation--Mathematicsmaster thesis10.11575/PRISM/38874