2 Layer Finite Difference Scheme PDE












0














I have this PDE, and I need to build a 2 layer Finite Difference scheme for it.



$frac{∂^2}{∂x^2}(k(x,t) frac{∂^2U(x,t)}{∂t^2})=0$



k is just a parameter, which is dependent on x and t. The problem is there are second order derivatives, and I don't know how to couple them together.










share|cite|improve this question









New contributor




P. Yastrebov is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

























    0














    I have this PDE, and I need to build a 2 layer Finite Difference scheme for it.



    $frac{∂^2}{∂x^2}(k(x,t) frac{∂^2U(x,t)}{∂t^2})=0$



    k is just a parameter, which is dependent on x and t. The problem is there are second order derivatives, and I don't know how to couple them together.










    share|cite|improve this question









    New contributor




    P. Yastrebov is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.























      0












      0








      0







      I have this PDE, and I need to build a 2 layer Finite Difference scheme for it.



      $frac{∂^2}{∂x^2}(k(x,t) frac{∂^2U(x,t)}{∂t^2})=0$



      k is just a parameter, which is dependent on x and t. The problem is there are second order derivatives, and I don't know how to couple them together.










      share|cite|improve this question









      New contributor




      P. Yastrebov is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.











      I have this PDE, and I need to build a 2 layer Finite Difference scheme for it.



      $frac{∂^2}{∂x^2}(k(x,t) frac{∂^2U(x,t)}{∂t^2})=0$



      k is just a parameter, which is dependent on x and t. The problem is there are second order derivatives, and I don't know how to couple them together.







      pde partial-derivative boundary-value-problem finite-difference-methods






      share|cite|improve this question









      New contributor




      P. Yastrebov is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.











      share|cite|improve this question









      New contributor




      P. Yastrebov is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.









      share|cite|improve this question




      share|cite|improve this question








      edited Dec 26 at 0:54





















      New contributor




      P. Yastrebov is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.









      asked Dec 26 at 0:49









      P. Yastrebov

      32




      32




      New contributor




      P. Yastrebov is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.





      New contributor





      P. Yastrebov is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.






      P. Yastrebov is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.






















          1 Answer
          1






          active

          oldest

          votes


















          0














          First we need to define a grid for the finite difference scheme. I will be using a regular gird here, but the same approach can be used for an irregular grid if the appropriate approximations to derivatives are used. The grid is
          $$
          x_i = x_0 + icdotDelta x
          quadquad
          t_j = t_0 + jcdotDelta t
          $$

          and for a general funciton $f(x,t)$, define
          $$
          f_{i,j} = f(x_i,t_j)
          $$

          which, on a uniform grid, has approximations to its (unmixed) second derivatives
          $$
          left(frac{partial^2 f}{partial x^2}right)_{i,j} = frac{f_{i+1,j} - 2f_{i,j} + f_{i-1,j}}{Delta x^2}
          $$

          $$
          left(frac{partial^2 f}{partial t^2}right)_{i,j} = frac{f_{i,j+1} - 2f_{i,j} + f_{i,j-1}}{Delta t^2}.
          $$



          For your given equation, we first define
          $$
          F(x,t) = k(x,t) cdot frac{partial^2 U}{partial t^2}(x,t)
          $$

          thus
          $$
          0 = frac{partial^2 F}{partial x^2}(x,t).
          $$

          Discretising:
          begin{align}
          0
          &= frac{1}{Delta x^2}left[F_{i+1,j} - 2F_{i,j} + F_{i-1,j}right]
          \
          &= frac{1}{Delta x^2}left[k_{i+1,j}left(frac{partial^2 U}{partial t^2}right)_{i+1,j} - 2k_{i,j}left(frac{partial^2 U}{partial t^2}right)_{i,j} + k_{i-1,j}left(frac{partial^2 U}{partial t^2}right)_{i-1,j}right]
          \
          &= frac{1}{Delta x^2Delta t^2}left[k_{i+1,j}left(U_{i+1,j+1}-2U_{i+1,j}+U_{i+1,j-1}right) - 2k_{i,j}left(U_{i,j+1}-2U_{i,j}+U_{i,j-1}right) +
          k_{i-1,j}left(U_{i-1,j+1}-2U_{i-1,j}+U_{i-1,j-1}right)right].
          end{align}

          To use this in a finite difference scheme you will need to specify $U_{i,j}$ at the first two time instances, as well as the left and right endpoints.






          share|cite|improve this answer





















            Your Answer





            StackExchange.ifUsing("editor", function () {
            return StackExchange.using("mathjaxEditing", function () {
            StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
            StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
            });
            });
            }, "mathjax-editing");

            StackExchange.ready(function() {
            var channelOptions = {
            tags: "".split(" "),
            id: "69"
            };
            initTagRenderer("".split(" "), "".split(" "), channelOptions);

            StackExchange.using("externalEditor", function() {
            // Have to fire editor after snippets, if snippets enabled
            if (StackExchange.settings.snippets.snippetsEnabled) {
            StackExchange.using("snippets", function() {
            createEditor();
            });
            }
            else {
            createEditor();
            }
            });

            function createEditor() {
            StackExchange.prepareEditor({
            heartbeatType: 'answer',
            autoActivateHeartbeat: false,
            convertImagesToLinks: true,
            noModals: true,
            showLowRepImageUploadWarning: true,
            reputationToPostImages: 10,
            bindNavPrevention: true,
            postfix: "",
            imageUploader: {
            brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
            contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
            allowUrls: true
            },
            noCode: true, onDemand: true,
            discardSelector: ".discard-answer"
            ,immediatelyShowMarkdownHelp:true
            });


            }
            });






            P. Yastrebov is a new contributor. Be nice, and check out our Code of Conduct.










            draft saved

            draft discarded


















            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3052546%2f2-layer-finite-difference-scheme-pde%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown

























            1 Answer
            1






            active

            oldest

            votes








            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            0














            First we need to define a grid for the finite difference scheme. I will be using a regular gird here, but the same approach can be used for an irregular grid if the appropriate approximations to derivatives are used. The grid is
            $$
            x_i = x_0 + icdotDelta x
            quadquad
            t_j = t_0 + jcdotDelta t
            $$

            and for a general funciton $f(x,t)$, define
            $$
            f_{i,j} = f(x_i,t_j)
            $$

            which, on a uniform grid, has approximations to its (unmixed) second derivatives
            $$
            left(frac{partial^2 f}{partial x^2}right)_{i,j} = frac{f_{i+1,j} - 2f_{i,j} + f_{i-1,j}}{Delta x^2}
            $$

            $$
            left(frac{partial^2 f}{partial t^2}right)_{i,j} = frac{f_{i,j+1} - 2f_{i,j} + f_{i,j-1}}{Delta t^2}.
            $$



            For your given equation, we first define
            $$
            F(x,t) = k(x,t) cdot frac{partial^2 U}{partial t^2}(x,t)
            $$

            thus
            $$
            0 = frac{partial^2 F}{partial x^2}(x,t).
            $$

            Discretising:
            begin{align}
            0
            &= frac{1}{Delta x^2}left[F_{i+1,j} - 2F_{i,j} + F_{i-1,j}right]
            \
            &= frac{1}{Delta x^2}left[k_{i+1,j}left(frac{partial^2 U}{partial t^2}right)_{i+1,j} - 2k_{i,j}left(frac{partial^2 U}{partial t^2}right)_{i,j} + k_{i-1,j}left(frac{partial^2 U}{partial t^2}right)_{i-1,j}right]
            \
            &= frac{1}{Delta x^2Delta t^2}left[k_{i+1,j}left(U_{i+1,j+1}-2U_{i+1,j}+U_{i+1,j-1}right) - 2k_{i,j}left(U_{i,j+1}-2U_{i,j}+U_{i,j-1}right) +
            k_{i-1,j}left(U_{i-1,j+1}-2U_{i-1,j}+U_{i-1,j-1}right)right].
            end{align}

            To use this in a finite difference scheme you will need to specify $U_{i,j}$ at the first two time instances, as well as the left and right endpoints.






            share|cite|improve this answer


























              0














              First we need to define a grid for the finite difference scheme. I will be using a regular gird here, but the same approach can be used for an irregular grid if the appropriate approximations to derivatives are used. The grid is
              $$
              x_i = x_0 + icdotDelta x
              quadquad
              t_j = t_0 + jcdotDelta t
              $$

              and for a general funciton $f(x,t)$, define
              $$
              f_{i,j} = f(x_i,t_j)
              $$

              which, on a uniform grid, has approximations to its (unmixed) second derivatives
              $$
              left(frac{partial^2 f}{partial x^2}right)_{i,j} = frac{f_{i+1,j} - 2f_{i,j} + f_{i-1,j}}{Delta x^2}
              $$

              $$
              left(frac{partial^2 f}{partial t^2}right)_{i,j} = frac{f_{i,j+1} - 2f_{i,j} + f_{i,j-1}}{Delta t^2}.
              $$



              For your given equation, we first define
              $$
              F(x,t) = k(x,t) cdot frac{partial^2 U}{partial t^2}(x,t)
              $$

              thus
              $$
              0 = frac{partial^2 F}{partial x^2}(x,t).
              $$

              Discretising:
              begin{align}
              0
              &= frac{1}{Delta x^2}left[F_{i+1,j} - 2F_{i,j} + F_{i-1,j}right]
              \
              &= frac{1}{Delta x^2}left[k_{i+1,j}left(frac{partial^2 U}{partial t^2}right)_{i+1,j} - 2k_{i,j}left(frac{partial^2 U}{partial t^2}right)_{i,j} + k_{i-1,j}left(frac{partial^2 U}{partial t^2}right)_{i-1,j}right]
              \
              &= frac{1}{Delta x^2Delta t^2}left[k_{i+1,j}left(U_{i+1,j+1}-2U_{i+1,j}+U_{i+1,j-1}right) - 2k_{i,j}left(U_{i,j+1}-2U_{i,j}+U_{i,j-1}right) +
              k_{i-1,j}left(U_{i-1,j+1}-2U_{i-1,j}+U_{i-1,j-1}right)right].
              end{align}

              To use this in a finite difference scheme you will need to specify $U_{i,j}$ at the first two time instances, as well as the left and right endpoints.






              share|cite|improve this answer
























                0












                0








                0






                First we need to define a grid for the finite difference scheme. I will be using a regular gird here, but the same approach can be used for an irregular grid if the appropriate approximations to derivatives are used. The grid is
                $$
                x_i = x_0 + icdotDelta x
                quadquad
                t_j = t_0 + jcdotDelta t
                $$

                and for a general funciton $f(x,t)$, define
                $$
                f_{i,j} = f(x_i,t_j)
                $$

                which, on a uniform grid, has approximations to its (unmixed) second derivatives
                $$
                left(frac{partial^2 f}{partial x^2}right)_{i,j} = frac{f_{i+1,j} - 2f_{i,j} + f_{i-1,j}}{Delta x^2}
                $$

                $$
                left(frac{partial^2 f}{partial t^2}right)_{i,j} = frac{f_{i,j+1} - 2f_{i,j} + f_{i,j-1}}{Delta t^2}.
                $$



                For your given equation, we first define
                $$
                F(x,t) = k(x,t) cdot frac{partial^2 U}{partial t^2}(x,t)
                $$

                thus
                $$
                0 = frac{partial^2 F}{partial x^2}(x,t).
                $$

                Discretising:
                begin{align}
                0
                &= frac{1}{Delta x^2}left[F_{i+1,j} - 2F_{i,j} + F_{i-1,j}right]
                \
                &= frac{1}{Delta x^2}left[k_{i+1,j}left(frac{partial^2 U}{partial t^2}right)_{i+1,j} - 2k_{i,j}left(frac{partial^2 U}{partial t^2}right)_{i,j} + k_{i-1,j}left(frac{partial^2 U}{partial t^2}right)_{i-1,j}right]
                \
                &= frac{1}{Delta x^2Delta t^2}left[k_{i+1,j}left(U_{i+1,j+1}-2U_{i+1,j}+U_{i+1,j-1}right) - 2k_{i,j}left(U_{i,j+1}-2U_{i,j}+U_{i,j-1}right) +
                k_{i-1,j}left(U_{i-1,j+1}-2U_{i-1,j}+U_{i-1,j-1}right)right].
                end{align}

                To use this in a finite difference scheme you will need to specify $U_{i,j}$ at the first two time instances, as well as the left and right endpoints.






                share|cite|improve this answer












                First we need to define a grid for the finite difference scheme. I will be using a regular gird here, but the same approach can be used for an irregular grid if the appropriate approximations to derivatives are used. The grid is
                $$
                x_i = x_0 + icdotDelta x
                quadquad
                t_j = t_0 + jcdotDelta t
                $$

                and for a general funciton $f(x,t)$, define
                $$
                f_{i,j} = f(x_i,t_j)
                $$

                which, on a uniform grid, has approximations to its (unmixed) second derivatives
                $$
                left(frac{partial^2 f}{partial x^2}right)_{i,j} = frac{f_{i+1,j} - 2f_{i,j} + f_{i-1,j}}{Delta x^2}
                $$

                $$
                left(frac{partial^2 f}{partial t^2}right)_{i,j} = frac{f_{i,j+1} - 2f_{i,j} + f_{i,j-1}}{Delta t^2}.
                $$



                For your given equation, we first define
                $$
                F(x,t) = k(x,t) cdot frac{partial^2 U}{partial t^2}(x,t)
                $$

                thus
                $$
                0 = frac{partial^2 F}{partial x^2}(x,t).
                $$

                Discretising:
                begin{align}
                0
                &= frac{1}{Delta x^2}left[F_{i+1,j} - 2F_{i,j} + F_{i-1,j}right]
                \
                &= frac{1}{Delta x^2}left[k_{i+1,j}left(frac{partial^2 U}{partial t^2}right)_{i+1,j} - 2k_{i,j}left(frac{partial^2 U}{partial t^2}right)_{i,j} + k_{i-1,j}left(frac{partial^2 U}{partial t^2}right)_{i-1,j}right]
                \
                &= frac{1}{Delta x^2Delta t^2}left[k_{i+1,j}left(U_{i+1,j+1}-2U_{i+1,j}+U_{i+1,j-1}right) - 2k_{i,j}left(U_{i,j+1}-2U_{i,j}+U_{i,j-1}right) +
                k_{i-1,j}left(U_{i-1,j+1}-2U_{i-1,j}+U_{i-1,j-1}right)right].
                end{align}

                To use this in a finite difference scheme you will need to specify $U_{i,j}$ at the first two time instances, as well as the left and right endpoints.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered 2 days ago









                Eddy

                774412




                774412






















                    P. Yastrebov is a new contributor. Be nice, and check out our Code of Conduct.










                    draft saved

                    draft discarded


















                    P. Yastrebov is a new contributor. Be nice, and check out our Code of Conduct.













                    P. Yastrebov is a new contributor. Be nice, and check out our Code of Conduct.












                    P. Yastrebov is a new contributor. Be nice, and check out our Code of Conduct.
















                    Thanks for contributing an answer to Mathematics Stack Exchange!


                    • Please be sure to answer the question. Provide details and share your research!

                    But avoid



                    • Asking for help, clarification, or responding to other answers.

                    • Making statements based on opinion; back them up with references or personal experience.


                    Use MathJax to format equations. MathJax reference.


                    To learn more, see our tips on writing great answers.





                    Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


                    Please pay close attention to the following guidance:


                    • Please be sure to answer the question. Provide details and share your research!

                    But avoid



                    • Asking for help, clarification, or responding to other answers.

                    • Making statements based on opinion; back them up with references or personal experience.


                    To learn more, see our tips on writing great answers.




                    draft saved


                    draft discarded














                    StackExchange.ready(
                    function () {
                    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3052546%2f2-layer-finite-difference-scheme-pde%23new-answer', 'question_page');
                    }
                    );

                    Post as a guest















                    Required, but never shown





















































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown

































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown







                    Popular posts from this blog

                    Human spaceflight

                    Can not write log (Is /dev/pts mounted?) - openpty in Ubuntu-on-Windows?

                    張江高科駅