This　paper　introduces　an　advanced　localized　space-time　Trefftz　method　tackling　boundary　value　predicaments　within　complex　two-dimensional　domains　governed　by　diffusion　equations.　In　contrast　to　the　widespread　space-time　collocation　Trefftz　method,　which　typically　produces　dense　and　ill-conditioned　matrices,　the　proposed　strategy　employs　a　localized　collocation　scheme　to　remove　these　constraints.　In　particular,　this　is　beneficial　in　multi-connected　configurations　or　when　dealing　with　significant　variations　in　field　values.　To　the　best　of　our　knowledge,　this　is　the　first　space-time　collocation　Trefftz　method　adaptation,　which　is　referred　to　as　the　localized　space-time　Trefftz　method　in　this　paper.　The　latter　combines　the　space-time　collocation　Trefftz　method　principles,　which　allows　to　eliminate　the　need　for　mesh　and　numerical　quadrature　in　its　application.　The　localized　space-time　Trefftz　method　represents　each　interior　node　expressed　as　a　linear　blend　of　its　immediate　neighbors,　while　the　space-time　collocation　Trefftz　method　applies　numerical　techniques　within　distinct　subdomains.　A　sparse　system　of　linear　algebraic　equations　with　internal　points　satisfying　the　governing　equation,　and　boundary　points　satisfying　the　boundary　conditions,　allows　to　obtain　numerical　solutions　using　matrix　systems.　The　localized　space-time　Trefftz　method　retains　the　easy-to-use　properties　and　mesh-free　structure　of　the　space-time　collocation　Trefftz　method,　and　it　mitigates　its　ill-conditioning　characteristics.　Due　to　the　localization　principle　and　the　consideration　of　overlapping　subdomains,　the　solutions　in　the　proposed　localized　space-time　Trefftz　method　are　more　simply　and　compactly　expressed　compared　with　those　in　the　space-time　collocation　Trefftz　method,　especially　when　dealing　with　multiply-connected　domains.　Numerical　examples　for　simply-connected　and　multiply-connected　domains　are　then　provided　to　demonstrate　the　high　precision　and　simplicity　of　the　proposed　localized　space-time　Trefftz　method.　The　obtained　results　show　that　the　latter　has　very　high　accuracy　in　solving　two-dimensional　diffusion　problems.　Compared　with　the　traditional　space-time　collocation　Trefftz　method,　the　proposed　mesh-free　strategy　yields　solutions　with　higher　precision　while　significantly　reducing　the　instability.　©　2024