{
  "id": "2102.05910",
  "version": "v1",
  "published": "2021-02-11T09:41:42.000Z",
  "updated": "2021-02-11T09:41:42.000Z",
  "title": "Higher-order generalized-$α$ methods for parabolic problems",
  "authors": [
    "Pouria Behnoudfar",
    "Quanling Deng",
    "Victor M. Calo"
  ],
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "We propose a new class of high-order time-marching schemes with dissipation user-control and unconditional stability for parabolic equations. High-order time integrators can deliver the optimal performance of highly-accurate and robust spatial discretizations such as isogeometric analysis. The generalized-$\\alpha$ method delivers unconditional stability and second-order accuracy in time and controls the numerical dissipation in the discrete spectrum's high-frequency region. Our goal is to extend the generalized-$alpha$ methodology to obtain a high-order time marching methods with high accuracy and dissipation in the discrete high-frequency range. Furthermore, we maintain the stability region of the original, second-order generalized-$alpha$ method foe the new higher-order methods. That is, we increase the accuracy of the generalized-$\\alpha$ method while keeping the unconditional stability and user-control features on the high-frequency numerical dissipation. The methodology solve $k>1, k\\in \\mathbb{N}$ matrix problems and updates the system unknowns, which correspond to higher-order terms in Taylor expansions to obtain $(3/2k)^{th}$-order method for even $k$ and $(3/2k+1/2)^{th}$-order for odd $k$. A single parameter $\\rho^\\infty$ controls the dissipation, and the update procedure follows the formulation of the original second-order method. Additionally, we show that our method is A-stable and setting $\\rho^\\infty=0$ allows us to obtain an L-stable method. Lastly, we extend this strategy to analyze the accuracy order of a generic method.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-11T09:41:42.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "parabolic problems",
      "higher-order",
      "method delivers unconditional stability",
      "discrete spectrums high-frequency region",
      "numerical dissipation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}