Fractional diffusion equations are generalizations of classical diffusion equations which are used in modeling practical superdiffusive problems in fluid flow, finance and others. Because of the nonlocal property of fractional differential operators, the numerical methods for fractional diffusion equations often generate dense or even full coefficient matrices, which results in computational work of O(N^3) per time step and memory of O(N^2) where N is the number of grid points. While a lot of methods are dealing with one dimensional problem, in this paper we develop a multigrid acceleration for spacial fractional diffusion equations in two space dimensions. The method only requires computational work of O(N logN) and memory of O(N) per time step, while retaining the same accuracy and approximation property as the regular finite difference method with Gaussian elimination. Our preliminary numerical examples run for two dimensional model problem of intermediate size seem to indicate the observations: To achieve the same accuracy, the new method has a significant reduction of the CPU time from more than 1 months consumed by a traditional finite difference method to less than 20 min, using less than one thousandth of memory the standard method does. This demonstrates the utility of the method. iv
","abstract_html":"<p>Fractional diffusion equations are generalizations of classical diffusion equations which are used in modeling practical superdiffusive problems in fluid flow, finance and others. Because of the nonlocal property of fractional differential operators, the numerical methods for fractional diffusion equations often generate dense or even full coefficient matrices, which results in computational work of O(N^3) per time step and memory of O(N^2) where N is the number of grid points. While a lot of methods are dealing with one dimensional problem, in this paper we develop a multigrid acceleration for spacial fractional diffusion equations in two space dimensions. The method only requires computational work of O(N logN) and memory of O(N) per time step, while retaining the same accuracy and approximation property as the regular finite difference method with Gaussian elimination. Our preliminary numerical examples run for two dimensional model problem of intermediate size seem to indicate the observations: To achieve the same accuracy, the new method has a significant reduction of the CPU time from more than 1 months consumed by a traditional finite difference method to less than 20 min, using less than one thousandth of memory the standard method does. This demonstrates the utility of the method. iv</p>","abstract_has_math":false,"creators":["Wang, Liwei"],"institution":null,"degree_name":"M.S.","degree_level":"Campus Access Thesis","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Hong Wang"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2012,"date_issued":"2012-01-01T08:00:00Z","date_published":"2012-01-01T08:00:00Z","updated_at":"2026-07-24T04:38:31Z","subjects":["Mathematics","Physical Sciences and Mathematics","alternating direct implicit method","fast fourier transform","fractional differential equations","multigrid","Toeplitz matrix"],"languages":[],"rights":["© 2012, Liwei Wang"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://scholarcommons.sc.edu/etd/1615","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Hong Wang"]},{"key":"dc:creator","label":"Author","values":["Wang, Liwei"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Campus Access Thesis"]},{"key":"thesis:degree_name","label":"Degree Name","values":["M.S."]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics","Physical Sciences and Mathematics","alternating direct implicit method","fast fourier transform","fractional differential equations","multigrid","Toeplitz matrix"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["© 2012, Liwei Wang"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://scholarcommons.sc.edu/etd/1615"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Fractional diffusion equations are generalizations of classical diffusion equations which are used in modeling practical superdiffusive problems in fluid flow, finance and others. Because of the nonlocal property of fractional differential operators, the numerical methods for fractional diffusion equations often generate dense or even full coefficient matrices, which results in computational work of O(N^3) per time step and memory of O(N^2) where N is the number of grid points. While a lot of methods are dealing with one dimensional problem, in this paper we develop a multigrid acceleration for spacial fractional diffusion equations in two space dimensions. The method only requires computational work of O(N logN) and memory of O(N) per time step, while retaining the same accuracy and approximation property as the regular finite difference method with Gaussian elimination. Our preliminary numerical examples run for two dimensional model problem of intermediate size seem to indicate the observations: To achieve the same accuracy, the new method has a significant reduction of the CPU time from more than 1 months consumed by a traditional finite difference method to less than 20 min, using less than one thousandth of memory the standard method does. This demonstrates the utility of the method. iv
"]},{"key":"dc:title","label":"Title","values":["A Multigrid Acceleration for an ADI Finite Difference Method for Two-Dimensional Space Fractional Diffusion Equations"]}]}],"canonical_facts":{"dc:contributor":["Hong Wang"],"dc:creator":["Wang, Liwei"],"dc:description.abstract":["Fractional diffusion equations are generalizations of classical diffusion equations which are used in modeling practical superdiffusive problems in fluid flow, finance and others. Because of the nonlocal property of fractional differential operators, the numerical methods for fractional diffusion equations often generate dense or even full coefficient matrices, which results in computational work of O(N^3) per time step and memory of O(N^2) where N is the number of grid points. While a lot of methods are dealing with one dimensional problem, in this paper we develop a multigrid acceleration for spacial fractional diffusion equations in two space dimensions. The method only requires computational work of O(N logN) and memory of O(N) per time step, while retaining the same accuracy and approximation property as the regular finite difference method with Gaussian elimination. Our preliminary numerical examples run for two dimensional model problem of intermediate size seem to indicate the observations: To achieve the same accuracy, the new method has a significant reduction of the CPU time from more than 1 months consumed by a traditional finite difference method to less than 20 min, using less than one thousandth of memory the standard method does. This demonstrates the utility of the method. iv
"],"dc:identifier":["https://scholarcommons.sc.edu/etd/1615"],"dc:rights":["© 2012, Liwei Wang"],"dc:subject":["Mathematics","Physical Sciences and Mathematics","alternating direct implicit method","fast fourier transform","fractional differential equations","multigrid","Toeplitz matrix"],"dc:title":["A Multigrid Acceleration for an ADI Finite Difference Method for Two-Dimensional Space Fractional Diffusion Equations"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Campus Access Thesis"],"thesis:degree_name":["M.S."]},"updated_at":"2026-07-24T04:38:31Z"}