{"paper":{"title":"Discretization of the Burgers' equation as a port-Hamiltonian system","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"A dedicated finite element method discretizes the Burgers equation into a finite-dimensional port-Hamiltonian system.","cross_cats":["cs.NA","math.AP"],"primary_cat":"math.NA","authors_text":"Ghislain Haine, Lorenzo Agostini, Michel Fourni\\'e","submitted_at":"2026-03-13T13:49:21Z","abstract_excerpt":"The numerical simulation of the inviscid Burgers' equation is often hindered by spurious oscillations near discontinuities. To mitigate this issue, a viscous term can be introduced, leading to the viscous Burgers' equation. In this work, port-Hamiltonian formulations for both the inviscid and the viscous Burgers' equations are proposed, enabling a representation that incorporates both convective and dissipative effects. Boundary control and observation are naturally handled within this framework. Applying a dedicated finite element method, a finite-dimensional port-Hamiltonian system is derive"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Applying a dedicated finite element method, a finite-dimensional port-Hamiltonian system is derived. The relationship between time step, spatial resolution, and viscosity required to achieve numerical stability is analyzed. Numerical experiments validate the effectiveness of the approach.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the chosen finite-element spaces and time-stepping scheme preserve the port-Hamiltonian structure and that the derived stability relation between time step, mesh size, and viscosity is both necessary and sufficient for the nonlinear problem.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Port-Hamiltonian discretizations of the Burgers' equation are derived via finite elements, yielding structure-preserving schemes whose stability conditions are analyzed and tested numerically.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A dedicated finite element method discretizes the Burgers equation into a finite-dimensional port-Hamiltonian system.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"95e6495f0be1f448c53501126ce679ad1c0e5b6063d37e95975996e883ec09d1"},"source":{"id":"2603.12992","kind":"arxiv","version":2},"verdict":{"id":"f365a004-a913-4f11-aa80-7e2303ec8d39","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T11:40:42.301060Z","strongest_claim":"Applying a dedicated finite element method, a finite-dimensional port-Hamiltonian system is derived. The relationship between time step, spatial resolution, and viscosity required to achieve numerical stability is analyzed. Numerical experiments validate the effectiveness of the approach.","one_line_summary":"Port-Hamiltonian discretizations of the Burgers' equation are derived via finite elements, yielding structure-preserving schemes whose stability conditions are analyzed and tested numerically.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the chosen finite-element spaces and time-stepping scheme preserve the port-Hamiltonian structure and that the derived stability relation between time step, mesh size, and viscosity is both necessary and sufficient for the nonlinear problem.","pith_extraction_headline":"A dedicated finite element method discretizes the Burgers equation into a finite-dimensional port-Hamiltonian system."},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}