[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84013-en":3,"doc-seo-84013-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},84013,7971461740909,"Levi","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","The WaveHoltz Heterogeneous Multiscale Method","Numerical methods for solving the wave equation in media with rapidly varying coefficients under time-harmonic sources are addressed using multiscale discretizations with prohibitive direct cost avoided through homogenization and frequency-lean strategies. The approach combines a finite-difference heterogeneous multiscale method (HMM) in the time domain with the WaveHoltz procedure that advances toward a time-periodic Helmholtz solution. Advantages of WaveHoltz over traditional Helmholtz solvers extend to multiscale settings, eliminating micro-scale boundary artifacts in the homogenized frequency-domain result.","arXiv :2607 .05811v1 [math .NA] 7 Jul 2026  \nTHE WAVEHOLTZ HETEROGENEOUS MULTISCALE METHOD. ∗  \nAMIT ROTEM†, OLOF RUNBORG‡, AND DANIEL APPEL¨O§  \nAbstract. We consider the numerical solution of the wave equation in materials with rapidly varying coefficients, and time harmonic sources. For these problems, direct discretization is prohibitively costly, and instead multiscale methods are used. There are several multiscale methods that directly discretize in the frequency domain. In this work we instead start in the time-domain and combine a finite difference Heterogeneous Multiscale Method (HMM) for the wave equation with the WaveHoltz method. Each WaveHoltz iteration marches the wave equation towards the time-periodic Helmholtz solution. The advantages of the WaveHoltz method relative to traditional Helmholtz solvers carry over directly to the multiscale problems considered here. Since, in addition, the time-domain solver does not artificially impose boundary conditions on the micro-scale problems, no boundary errors from the micro-scale problems are present in the homogenized frequency domain solution.  \nKey words. WaveHoltz, heterogeneous multiscale method, Helmholtz, waves, homogenization. MSC codes. 35J05, 65N06, 65N12, 74Q10  \n1. Introduction. This paper considers the Helmholtz equation,(1 . 1) ∇ · (aε (x)∇u) + ω2 u = b (x), x ∈ Ω ,  \nwith Dirichlet or Neumann boundary conditions. Here the computational domain Ω ⊂ Rd (d = 1 or 2) is a smooth, bounded, and simply connected domain, b ∈ L2 (Ω) is independent of ε and the Helmholtz frequency ω is assumed to be nonresonant. The scalar valued function aε (x) is uniformly bounded and positive. Weare interested in the case where aε is rapidly varying, i.e. its smallest scale ε is much smaller than the longest wavelengths ∝ ∥ √aε ∥L∞ /ω of the solution. Our goal is to compute approximations to the solution u for problems with scale separation. By scale separation we mean that the solution u(x) can be expanded in powers of ε as  \nu (x) = u¯(x) + εu1 (x, x/ε) + O(ε2 ) ,  \nwhere, the zeroth order term u¯ does not depend on ε . In the particular case when aε (x) = a(x, x/ε) is periodic in the fast variable x/ε, the zeroth order term u¯ can be shown [34, 12, 6] to satisfy a homogenized equation  \n(1 .2) ∇ · (a¯(x)∇u¯) + ω 2 u¯ = b (x), x ∈ Ω .  \nNote that a¯ does not depend on ε and thus u¯ need only be resolved at the length scales of the homogenized coefficient a¯, the wavelengths associated with the frequency ω , and the scales of the forcing b. When these are large compared to ε and when ε is small enough for the homogenized solution u¯ to be a good approximation of u,  \n∗ Submitted to the editors  \nFunding: DA is supported by the U.S. Department of Energy, Office of Science, Advanced Scientific Computing Research (ASCR), under Award Number DE-SC0025424 . This material is based upon work supported, in part, by the National Science Foundation under Grant No. DMS- 2436319 and Virginia Tech. This material is based upon work supported by the National Science Foundation under Grant No. DMS-2424139 while the third author was in residence at the Simons Laufer Mathematical Sciences Institute in Berkeley, California, during the Fall 2025 semester.  \n†Department of Mathematics, Virginia Tech ([arotem@vt.edu](arotem@vt.edu)) .  \n‡Department of Mathematics, KTH Royal Institute of Technology ([olofr@kth.se](olofr@kth.se)) .  \n§ Department of Mathematics, Virginia Tech ([appelo@vt.edu](appelo@vt.edu)) .  \n2 A. ROTEM, O. RUNBORG, AND D. APPEL¨O  \nhomogenization provides an accurate approximation to u at a low computational cost.  \nHowever, solving or discretizing (1.2) requires explicit knowledge of a¯ . When aε is periodic and depends only on the fast scale, i.e. aε (x) = a (x/ε), the homogenized coefficient a¯ can be computed by solving the so called cell problem, an associated elliptic problem with periodic boundary conditions, [12] . When aε is ε-periodic but also has a slowly varying componen","cbCaigcI6XLq17F8","https://ap.wps.com/l/cbCaigcI6XLq17F8","pdf",2447454,4,1,29,"English","en",105,"# Introduction\n## Literature Review","[{\"question\":\"What problem does the WaveHoltz heterogeneous multiscale method target?\",\"answer\":\"It targets numerical solutions of the wave equation in materials with rapidly varying coefficients and time-harmonic sources, where direct discretization is too costly.\"},{\"question\":\"How is the multiscale strategy constructed in this method?\",\"answer\":\"The method starts in the time domain by combining a finite-difference Heterogeneous Multiscale Method (HMM) with WaveHoltz iterations that march toward the time-periodic Helmholtz solution.\"},{\"question\":\"Why are micro-scale boundary errors avoided in the homogenized frequency-domain solution?\",\"answer\":\"Because the time-domain solver does not artificially impose boundary conditions on micro-scale problems, boundary errors from those problems do not appear in the homogenized frequency-domain 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problem does the WaveHoltz heterogeneous multiscale method target?","Question",{"text":75,"@type":76},"It targets numerical solutions of the wave equation in materials with rapidly varying coefficients and time-harmonic sources, where direct discretization is too costly.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is the multiscale strategy constructed in this method?",{"text":80,"@type":76},"The method starts in the time domain by combining a finite-difference Heterogeneous Multiscale Method (HMM) with WaveHoltz iterations that march toward the time-periodic Helmholtz solution.",{"name":82,"@type":73,"acceptedAnswer":83},"Why are micro-scale boundary errors avoided in the homogenized frequency-domain solution?",{"text":84,"@type":76},"Because the time-domain solver does not artificially impose boundary conditions on micro-scale problems, boundary errors from those problems do not appear in the homogenized frequency-domain 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