Analytical solution for PDE-system's IBVP to validate method of lines












0












$begingroup$


I am in search of initial-boundary value problems which are posed in the form of a system of coupled PDE's (not a single PDE) and for which an analytical solution exists at least for some specific initial-boundary conditions.



In particular I would be interested in a PDE's system of one dynamical ($partial_r$ on lhs) equation coupled to one constraint ($partial_t$ on lhs) equation.



The generic character of the question is due to the fact that l have not yet come acrross any PDE's system with the above characteristics. Being more specific would probably make the question overdetermined.



The ultimate purpose of my search is to use the analytical solutions as a numerical test for Mathematica Method of Lines code.



I would appreciate any recommendations.






pde numerical-methods boundary-value-problem







share|cite|improve this question











$endgroup$












  • $begingroup$
    The heat equation has various benchmark problems with known solutions.
    $endgroup$
    – User123456789
    Dec 20 '18 at 10:48










  • $begingroup$
    This question is an enhanced version of a question I deleted. The deleted question did not lead to any usefull results except the comment by Skip linking to this PDE's system.
    $endgroup$
    – dkstack
    Dec 20 '18 at 10:49










  • $begingroup$
    @User123456789 these benchmark problems concern systems of coupled PDE's or single equations?
    $endgroup$
    – dkstack
    Dec 20 '18 at 10:52










  • $begingroup$
    You are right, I misread. You can maybe think of Stokes' equation then
    $endgroup$
    – User123456789
    Dec 20 '18 at 10:55










  • $begingroup$
    @User123456789 I searched in wikipedia about Stoke's equation but found nothing about PDE's system with an exact solution. Is there any link about the Stoke's equation you refer to?
    $endgroup$
    – dkstack
    Dec 20 '18 at 11:12


















0












$begingroup$


I am in search of initial-boundary value problems which are posed in the form of a system of coupled PDE's (not a single PDE) and for which an analytical solution exists at least for some specific initial-boundary conditions.



In particular I would be interested in a PDE's system of one dynamical ($partial_r$ on lhs) equation coupled to one constraint ($partial_t$ on lhs) equation.



The generic character of the question is due to the fact that l have not yet come acrross any PDE's system with the above characteristics. Being more specific would probably make the question overdetermined.



The ultimate purpose of my search is to use the analytical solutions as a numerical test for Mathematica Method of Lines code.



I would appreciate any recommendations.






pde numerical-methods boundary-value-problem







share|cite|improve this question











$endgroup$












  • $begingroup$
    The heat equation has various benchmark problems with known solutions.
    $endgroup$
    – User123456789
    Dec 20 '18 at 10:48










  • $begingroup$
    This question is an enhanced version of a question I deleted. The deleted question did not lead to any usefull results except the comment by Skip linking to this PDE's system.
    $endgroup$
    – dkstack
    Dec 20 '18 at 10:49










  • $begingroup$
    @User123456789 these benchmark problems concern systems of coupled PDE's or single equations?
    $endgroup$
    – dkstack
    Dec 20 '18 at 10:52










  • $begingroup$
    You are right, I misread. You can maybe think of Stokes' equation then
    $endgroup$
    – User123456789
    Dec 20 '18 at 10:55










  • $begingroup$
    @User123456789 I searched in wikipedia about Stoke's equation but found nothing about PDE's system with an exact solution. Is there any link about the Stoke's equation you refer to?
    $endgroup$
    – dkstack
    Dec 20 '18 at 11:12
















0












0








0


0



$begingroup$


I am in search of initial-boundary value problems which are posed in the form of a system of coupled PDE's (not a single PDE) and for which an analytical solution exists at least for some specific initial-boundary conditions.



In particular I would be interested in a PDE's system of one dynamical ($partial_r$ on lhs) equation coupled to one constraint ($partial_t$ on lhs) equation.



The generic character of the question is due to the fact that l have not yet come acrross any PDE's system with the above characteristics. Being more specific would probably make the question overdetermined.



The ultimate purpose of my search is to use the analytical solutions as a numerical test for Mathematica Method of Lines code.



I would appreciate any recommendations.






pde numerical-methods boundary-value-problem







share|cite|improve this question











$endgroup$




I am in search of initial-boundary value problems which are posed in the form of a system of coupled PDE's (not a single PDE) and for which an analytical solution exists at least for some specific initial-boundary conditions.



In particular I would be interested in a PDE's system of one dynamical ($partial_r$ on lhs) equation coupled to one constraint ($partial_t$ on lhs) equation.



The generic character of the question is due to the fact that l have not yet come acrross any PDE's system with the above characteristics. Being more specific would probably make the question overdetermined.



The ultimate purpose of my search is to use the analytical solutions as a numerical test for Mathematica Method of Lines code.



I would appreciate any recommendations.






pde numerical-methods boundary-value-problem




pde numerical-methods boundary-value-problem






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 21 '18 at 11:21







dkstack

















asked Dec 20 '18 at 10:46









dkstackdkstack

204




204












  • $begingroup$
    The heat equation has various benchmark problems with known solutions.
    $endgroup$
    – User123456789
    Dec 20 '18 at 10:48










  • $begingroup$
    This question is an enhanced version of a question I deleted. The deleted question did not lead to any usefull results except the comment by Skip linking to this PDE's system.
    $endgroup$
    – dkstack
    Dec 20 '18 at 10:49










  • $begingroup$
    @User123456789 these benchmark problems concern systems of coupled PDE's or single equations?
    $endgroup$
    – dkstack
    Dec 20 '18 at 10:52










  • $begingroup$
    You are right, I misread. You can maybe think of Stokes' equation then
    $endgroup$
    – User123456789
    Dec 20 '18 at 10:55










  • $begingroup$
    @User123456789 I searched in wikipedia about Stoke's equation but found nothing about PDE's system with an exact solution. Is there any link about the Stoke's equation you refer to?
    $endgroup$
    – dkstack
    Dec 20 '18 at 11:12




















  • $begingroup$
    The heat equation has various benchmark problems with known solutions.
    $endgroup$
    – User123456789
    Dec 20 '18 at 10:48










  • $begingroup$
    This question is an enhanced version of a question I deleted. The deleted question did not lead to any usefull results except the comment by Skip linking to this PDE's system.
    $endgroup$
    – dkstack
    Dec 20 '18 at 10:49










  • $begingroup$
    @User123456789 these benchmark problems concern systems of coupled PDE's or single equations?
    $endgroup$
    – dkstack
    Dec 20 '18 at 10:52










  • $begingroup$
    You are right, I misread. You can maybe think of Stokes' equation then
    $endgroup$
    – User123456789
    Dec 20 '18 at 10:55










  • $begingroup$
    @User123456789 I searched in wikipedia about Stoke's equation but found nothing about PDE's system with an exact solution. Is there any link about the Stoke's equation you refer to?
    $endgroup$
    – dkstack
    Dec 20 '18 at 11:12


















$begingroup$
The heat equation has various benchmark problems with known solutions.
$endgroup$
– User123456789
Dec 20 '18 at 10:48




$begingroup$
The heat equation has various benchmark problems with known solutions.
$endgroup$
– User123456789
Dec 20 '18 at 10:48












$begingroup$
This question is an enhanced version of a question I deleted. The deleted question did not lead to any usefull results except the comment by Skip linking to this PDE's system.
$endgroup$
– dkstack
Dec 20 '18 at 10:49




$begingroup$
This question is an enhanced version of a question I deleted. The deleted question did not lead to any usefull results except the comment by Skip linking to this PDE's system.
$endgroup$
– dkstack
Dec 20 '18 at 10:49












$begingroup$
@User123456789 these benchmark problems concern systems of coupled PDE's or single equations?
$endgroup$
– dkstack
Dec 20 '18 at 10:52




$begingroup$
@User123456789 these benchmark problems concern systems of coupled PDE's or single equations?
$endgroup$
– dkstack
Dec 20 '18 at 10:52












$begingroup$
You are right, I misread. You can maybe think of Stokes' equation then
$endgroup$
– User123456789
Dec 20 '18 at 10:55




$begingroup$
You are right, I misread. You can maybe think of Stokes' equation then
$endgroup$
– User123456789
Dec 20 '18 at 10:55












$begingroup$
@User123456789 I searched in wikipedia about Stoke's equation but found nothing about PDE's system with an exact solution. Is there any link about the Stoke's equation you refer to?
$endgroup$
– dkstack
Dec 20 '18 at 11:12






$begingroup$
@User123456789 I searched in wikipedia about Stoke's equation but found nothing about PDE's system with an exact solution. Is there any link about the Stoke's equation you refer to?
$endgroup$
– dkstack
Dec 20 '18 at 11:12












2 Answers
2






active

oldest

votes


















1












$begingroup$

You could use the "method of manufactured solutions (MMS)", which is nicely described in



https://www.comsol.de/blogs/verify-simulations-with-the-method-of-manufactured-solutions/



Essentially, you take some system of PDE, summarized as $$F[u]=r_F,$$
where $[u]$ includes the partial derivatives of $u$ that are used, and right side $r_B$ which is to be chosen later. Denote the boundary conditions similarly as $$B[u]=r_B.$$ Now select some function $p$ that is in the class of admissible functions and perhaps contains qualities expected in a solution.



Then the test system is $$F[u]=F[p],~~ B[u]=B[p],$$ that is, the right sides are computed as $r_F=F[p]$ and $r_B=B[p]$, and $p$ is obviously the exact solution.




  • discussion of MMS in a very simple context, with accordingly simple example https://scicomp.stackexchange.com/q/30562/6839






share|cite|improve this answer











$endgroup$





















    1












    $begingroup$

    The simplest may be a 1D linear hyperbolic system of conservation laws $q_t + A q_x = 0$, such as the system of linear acoustics
    begin{aligned}
    p_t + u_0 p_x + K_0 u_x &= 0,\
    u_t + p_x/rho_0 + u_0 u_x &= 0,
    end{aligned}
    the system of linear elasticity, or the system of electromagnetism. In all cases, $q in Bbb R^2$ and $A$ is a matrix with real eigenvalues. Almost every initial-and-boundary-value problem for such systems can be solved via the method of characteristics.






    share|cite|improve this answer











    $endgroup$














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      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      1












      $begingroup$

      You could use the "method of manufactured solutions (MMS)", which is nicely described in



      https://www.comsol.de/blogs/verify-simulations-with-the-method-of-manufactured-solutions/



      Essentially, you take some system of PDE, summarized as $$F[u]=r_F,$$
      where $[u]$ includes the partial derivatives of $u$ that are used, and right side $r_B$ which is to be chosen later. Denote the boundary conditions similarly as $$B[u]=r_B.$$ Now select some function $p$ that is in the class of admissible functions and perhaps contains qualities expected in a solution.



      Then the test system is $$F[u]=F[p],~~ B[u]=B[p],$$ that is, the right sides are computed as $r_F=F[p]$ and $r_B=B[p]$, and $p$ is obviously the exact solution.




      • discussion of MMS in a very simple context, with accordingly simple example https://scicomp.stackexchange.com/q/30562/6839






      share|cite|improve this answer











      $endgroup$


















        1












        $begingroup$

        You could use the "method of manufactured solutions (MMS)", which is nicely described in



        https://www.comsol.de/blogs/verify-simulations-with-the-method-of-manufactured-solutions/



        Essentially, you take some system of PDE, summarized as $$F[u]=r_F,$$
        where $[u]$ includes the partial derivatives of $u$ that are used, and right side $r_B$ which is to be chosen later. Denote the boundary conditions similarly as $$B[u]=r_B.$$ Now select some function $p$ that is in the class of admissible functions and perhaps contains qualities expected in a solution.



        Then the test system is $$F[u]=F[p],~~ B[u]=B[p],$$ that is, the right sides are computed as $r_F=F[p]$ and $r_B=B[p]$, and $p$ is obviously the exact solution.




        • discussion of MMS in a very simple context, with accordingly simple example https://scicomp.stackexchange.com/q/30562/6839






        share|cite|improve this answer











        $endgroup$
















          1












          1








          1





          $begingroup$

          You could use the "method of manufactured solutions (MMS)", which is nicely described in



          https://www.comsol.de/blogs/verify-simulations-with-the-method-of-manufactured-solutions/



          Essentially, you take some system of PDE, summarized as $$F[u]=r_F,$$
          where $[u]$ includes the partial derivatives of $u$ that are used, and right side $r_B$ which is to be chosen later. Denote the boundary conditions similarly as $$B[u]=r_B.$$ Now select some function $p$ that is in the class of admissible functions and perhaps contains qualities expected in a solution.



          Then the test system is $$F[u]=F[p],~~ B[u]=B[p],$$ that is, the right sides are computed as $r_F=F[p]$ and $r_B=B[p]$, and $p$ is obviously the exact solution.




          • discussion of MMS in a very simple context, with accordingly simple example https://scicomp.stackexchange.com/q/30562/6839






          share|cite|improve this answer











          $endgroup$



          You could use the "method of manufactured solutions (MMS)", which is nicely described in



          https://www.comsol.de/blogs/verify-simulations-with-the-method-of-manufactured-solutions/



          Essentially, you take some system of PDE, summarized as $$F[u]=r_F,$$
          where $[u]$ includes the partial derivatives of $u$ that are used, and right side $r_B$ which is to be chosen later. Denote the boundary conditions similarly as $$B[u]=r_B.$$ Now select some function $p$ that is in the class of admissible functions and perhaps contains qualities expected in a solution.



          Then the test system is $$F[u]=F[p],~~ B[u]=B[p],$$ that is, the right sides are computed as $r_F=F[p]$ and $r_B=B[p]$, and $p$ is obviously the exact solution.




          • discussion of MMS in a very simple context, with accordingly simple example https://scicomp.stackexchange.com/q/30562/6839







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Dec 20 '18 at 21:54

























          answered Dec 20 '18 at 21:24









          LutzLLutzL

          60.3k42057




          60.3k42057























              1












              $begingroup$

              The simplest may be a 1D linear hyperbolic system of conservation laws $q_t + A q_x = 0$, such as the system of linear acoustics
              begin{aligned}
              p_t + u_0 p_x + K_0 u_x &= 0,\
              u_t + p_x/rho_0 + u_0 u_x &= 0,
              end{aligned}
              the system of linear elasticity, or the system of electromagnetism. In all cases, $q in Bbb R^2$ and $A$ is a matrix with real eigenvalues. Almost every initial-and-boundary-value problem for such systems can be solved via the method of characteristics.






              share|cite|improve this answer











              $endgroup$


















                1












                $begingroup$

                The simplest may be a 1D linear hyperbolic system of conservation laws $q_t + A q_x = 0$, such as the system of linear acoustics
                begin{aligned}
                p_t + u_0 p_x + K_0 u_x &= 0,\
                u_t + p_x/rho_0 + u_0 u_x &= 0,
                end{aligned}
                the system of linear elasticity, or the system of electromagnetism. In all cases, $q in Bbb R^2$ and $A$ is a matrix with real eigenvalues. Almost every initial-and-boundary-value problem for such systems can be solved via the method of characteristics.






                share|cite|improve this answer











                $endgroup$
















                  1












                  1








                  1





                  $begingroup$

                  The simplest may be a 1D linear hyperbolic system of conservation laws $q_t + A q_x = 0$, such as the system of linear acoustics
                  begin{aligned}
                  p_t + u_0 p_x + K_0 u_x &= 0,\
                  u_t + p_x/rho_0 + u_0 u_x &= 0,
                  end{aligned}
                  the system of linear elasticity, or the system of electromagnetism. In all cases, $q in Bbb R^2$ and $A$ is a matrix with real eigenvalues. Almost every initial-and-boundary-value problem for such systems can be solved via the method of characteristics.






                  share|cite|improve this answer











                  $endgroup$



                  The simplest may be a 1D linear hyperbolic system of conservation laws $q_t + A q_x = 0$, such as the system of linear acoustics
                  begin{aligned}
                  p_t + u_0 p_x + K_0 u_x &= 0,\
                  u_t + p_x/rho_0 + u_0 u_x &= 0,
                  end{aligned}
                  the system of linear elasticity, or the system of electromagnetism. In all cases, $q in Bbb R^2$ and $A$ is a matrix with real eigenvalues. Almost every initial-and-boundary-value problem for such systems can be solved via the method of characteristics.







                  share|cite|improve this answer














                  share|cite|improve this answer



                  share|cite|improve this answer








                  edited Dec 20 '18 at 13:58

























                  answered Dec 20 '18 at 13:48









                  Harry49Harry49

                  8,56431344




                  8,56431344






























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