Fractional Steps Scheme to Approximate the Phase Field Transition System Endowed with Inhomogeneous/Homogeneous Cauchy-Neumann/Neumann Boundary Conditions
Here, we consider the phase field transition system (a nonlinear system of parabolic type) introduced by Caginalp to distinguish between the phases of the material that are involved in the solidification process. We start by investigating the solvability of such boundary value problems in the class...
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MDPI AG
2023-11-01
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| author | Constantin Fetecau Costică Moroşanu Dorin-Cătălin Stoicescu |
| author_facet | Constantin Fetecau Costică Moroşanu Dorin-Cătălin Stoicescu |
| author_sort | Constantin Fetecau |
| collection | DOAJ |
| description | Here, we consider the phase field transition system (a nonlinear system of parabolic type) introduced by Caginalp to distinguish between the phases of the material that are involved in the solidification process. We start by investigating the solvability of such boundary value problems in the class <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>W</mi><mi>p</mi><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msubsup><mrow><mo>(</mo><mi>Q</mi><mo>)</mo></mrow><mo>×</mo><msubsup><mi>W</mi><mi>ν</mi><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msubsup><mrow><mo>(</mo><mi>Q</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. One proves the existence, the regularity, and the uniqueness of solutions, in the presence of the cubic nonlinearity type. On the basis of the convergence of an iterative scheme of the fractional steps type, a conceptual numerical algorithm, <b>alg-frac_sec-ord-varphi_PHT</b>, is elaborated in order to approximate the solution of the nonlinear parabolic problem. The advantage of such an approach is that the new method simplifies the numerical computations due to its decoupling feature. An example of the numerical implementation of the principal step in the conceptual algorithm is also reported. Some conclusions are given are also given as new directions to extend the results and methods presented in the present paper. |
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| last_indexed | 2024-03-08T21:00:36Z |
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| spelling | doaj.art-6d356c472c014867b42cbdfcedb37ee72023-12-22T13:53:14ZengMDPI AGAxioms2075-16802023-11-011212109810.3390/axioms12121098Fractional Steps Scheme to Approximate the Phase Field Transition System Endowed with Inhomogeneous/Homogeneous Cauchy-Neumann/Neumann Boundary ConditionsConstantin Fetecau 0Costică Moroşanu1Dorin-Cătălin Stoicescu2Academy of Romanian Scientists, 54 Splaiul Independentei, 050094 Bucharest, RomaniaDepartment of Mathematics, “Alexandru Ioan Cuza” University, Bd. Carol I, 11, 700506 Iaşi, RomaniaFaculty of Automatic Control and Computer Engineering, Technical University “Gheorghe Asachi” of Iaşi, Dimitrie Mangeron, Nr. 27, 700050 Iaşi, RomaniaHere, we consider the phase field transition system (a nonlinear system of parabolic type) introduced by Caginalp to distinguish between the phases of the material that are involved in the solidification process. We start by investigating the solvability of such boundary value problems in the class <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>W</mi><mi>p</mi><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msubsup><mrow><mo>(</mo><mi>Q</mi><mo>)</mo></mrow><mo>×</mo><msubsup><mi>W</mi><mi>ν</mi><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msubsup><mrow><mo>(</mo><mi>Q</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. One proves the existence, the regularity, and the uniqueness of solutions, in the presence of the cubic nonlinearity type. On the basis of the convergence of an iterative scheme of the fractional steps type, a conceptual numerical algorithm, <b>alg-frac_sec-ord-varphi_PHT</b>, is elaborated in order to approximate the solution of the nonlinear parabolic problem. The advantage of such an approach is that the new method simplifies the numerical computations due to its decoupling feature. An example of the numerical implementation of the principal step in the conceptual algorithm is also reported. Some conclusions are given are also given as new directions to extend the results and methods presented in the present paper.https://www.mdpi.com/2075-1680/12/12/1098boundary value problems for nonlinear parabolic PDEfractional steps methodconvergence of numerical schemenumerical algorithmphase-changes |
| spellingShingle | Constantin Fetecau Costică Moroşanu Dorin-Cătălin Stoicescu Fractional Steps Scheme to Approximate the Phase Field Transition System Endowed with Inhomogeneous/Homogeneous Cauchy-Neumann/Neumann Boundary Conditions Axioms boundary value problems for nonlinear parabolic PDE fractional steps method convergence of numerical scheme numerical algorithm phase-changes |
| title | Fractional Steps Scheme to Approximate the Phase Field Transition System Endowed with Inhomogeneous/Homogeneous Cauchy-Neumann/Neumann Boundary Conditions |
| title_full | Fractional Steps Scheme to Approximate the Phase Field Transition System Endowed with Inhomogeneous/Homogeneous Cauchy-Neumann/Neumann Boundary Conditions |
| title_fullStr | Fractional Steps Scheme to Approximate the Phase Field Transition System Endowed with Inhomogeneous/Homogeneous Cauchy-Neumann/Neumann Boundary Conditions |
| title_full_unstemmed | Fractional Steps Scheme to Approximate the Phase Field Transition System Endowed with Inhomogeneous/Homogeneous Cauchy-Neumann/Neumann Boundary Conditions |
| title_short | Fractional Steps Scheme to Approximate the Phase Field Transition System Endowed with Inhomogeneous/Homogeneous Cauchy-Neumann/Neumann Boundary Conditions |
| title_sort | fractional steps scheme to approximate the phase field transition system endowed with inhomogeneous homogeneous cauchy neumann neumann boundary conditions |
| topic | boundary value problems for nonlinear parabolic PDE fractional steps method convergence of numerical scheme numerical algorithm phase-changes |
| url | https://www.mdpi.com/2075-1680/12/12/1098 |
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