Riemann problem of nonconvex scalar conservation laws The 2019 Stack Overflow Developer Survey Results Are In Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Find weak solution to Riemann problem for conservation lawSketch solution of IVP for nonconvex scalar conservation lawRankine-Hugoniot jump condition for non-homogeneous conservation lawreversibility scalar conservation lawWhat is the use of the notion of consistency for Riemann solvers?Solve the ivp for a scalar conservation lawConservation of mass in hyperbolic PDE [reference request]Entropy solution to scalar conservation lawFind the weak solution of the conservation lawThe Rankine-Hugoniot jump conditions for conservation and balance lawsWeak solutions of initial value problem of conservation laws with $L^infty$ initial dataNonsmooth data in the conservation laws, their approximations and limitsFinding the time when the speed of discontinuity becomes time-dependent in traffic flow

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Riemann problem of nonconvex scalar conservation laws



The 2019 Stack Overflow Developer Survey Results Are In
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Find weak solution to Riemann problem for conservation lawSketch solution of IVP for nonconvex scalar conservation lawRankine-Hugoniot jump condition for non-homogeneous conservation lawreversibility scalar conservation lawWhat is the use of the notion of consistency for Riemann solvers?Solve the ivp for a scalar conservation lawConservation of mass in hyperbolic PDE [reference request]Entropy solution to scalar conservation lawFind the weak solution of the conservation lawThe Rankine-Hugoniot jump conditions for conservation and balance lawsWeak solutions of initial value problem of conservation laws with $L^infty$ initial dataNonsmooth data in the conservation laws, their approximations and limitsFinding the time when the speed of discontinuity becomes time-dependent in traffic flow










2












$begingroup$


Consider the scalar conservation law $partial_t u+partial_xf(u)=0$. Riemann problem means the initial data given by



beginequation
u_0=begincases
u_L, & x<0 \
u_R, & xgeq 0
endcases
endequation



When $f(x)$ is convex, I know the corresponding theory. What if $f$ is not convex, for example $f(u)=fracu^33$, how to solve it?










share|cite|improve this question











$endgroup$
















    2












    $begingroup$


    Consider the scalar conservation law $partial_t u+partial_xf(u)=0$. Riemann problem means the initial data given by



    beginequation
    u_0=begincases
    u_L, & x<0 \
    u_R, & xgeq 0
    endcases
    endequation



    When $f(x)$ is convex, I know the corresponding theory. What if $f$ is not convex, for example $f(u)=fracu^33$, how to solve it?










    share|cite|improve this question











    $endgroup$














      2












      2








      2





      $begingroup$


      Consider the scalar conservation law $partial_t u+partial_xf(u)=0$. Riemann problem means the initial data given by



      beginequation
      u_0=begincases
      u_L, & x<0 \
      u_R, & xgeq 0
      endcases
      endequation



      When $f(x)$ is convex, I know the corresponding theory. What if $f$ is not convex, for example $f(u)=fracu^33$, how to solve it?










      share|cite|improve this question











      $endgroup$




      Consider the scalar conservation law $partial_t u+partial_xf(u)=0$. Riemann problem means the initial data given by



      beginequation
      u_0=begincases
      u_L, & x<0 \
      u_R, & xgeq 0
      endcases
      endequation



      When $f(x)$ is convex, I know the corresponding theory. What if $f$ is not convex, for example $f(u)=fracu^33$, how to solve it?







      pde hyperbolic-equations






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Nov 27 '17 at 15:15









      Harry49

      8,76331346




      8,76331346










      asked Nov 27 '17 at 8:44









      Kira YamatoKira Yamato

      552513




      552513




















          1 Answer
          1






          active

          oldest

          votes


















          5












          $begingroup$

          The method is very similar to the convex case, e.g. Burgers' equation where $f(u) = frac12u^2$, but there are more possible types of waves. In facts, in addition to shock waves and rarefaction waves, there may be waves with both discontinuous and continuous parts. Moreover, the Lax entropy condition for shocks must be replaced by the more general Oleinik entropy condition.



          In the case where the flux $f$ is not convex, these are the possible types of waves:




          • shock waves. If the solution is a shock wave with expression
            $$
            u(x,t) =
            leftlbrace
            beginaligned
            &u_L & &textifquad x < s, t , ,\
            &u_R & &textifquad s, t < x , ,
            endaligned
            right.
            tag1
            $$

            then the speed of shock $s$ must satisfy the Rankine-Hugoniot jump condition
            $s = fracf(u_R)- f(u_L)u_R - u_L$. Moreover, the shock wave must satisfy the Oleinik entropy condition [1]
            $$
            fracf(u)- f(u_L)u - u_L geq s geq fracf(u_R)- f(u)u_R - u ,
            $$

            for all $u$ between $u_L$ and $u_R$. In the case where $f$ is convex, the slope of its chords can be compared with its derivative using convexity inequalities. Thus, the classical Lax entropy condition $f'(u_L)>s>f'(u_R)$ is recovered, where $f'$ denotes the derivative of $f$.


          • rarefaction waves. The derivation is similar to the convex case, starting with the self-similarity Ansatz $u(x,t) = v(xi)$ where $xi = x/t$, which gives $f'(v(xi)) = xi$. In the nonconvex case, the equation $f'(v(xi)) = xi$ may have multiple solutions $v(xi)$, and the correct one is deduced from the continuity conditions $v(f'(u_L)) = u_L$ and $v(f'(u_R)) = u_R$. Such a solution is given by
            $$
            u(x,t) =
            leftlbrace
            beginaligned
            &u_L & &textifquad x leq f'(u_L), t , ,\
            &(f')^-1(x/t) & &textifquad f'(u_L), t leq x leq f'(u_R), t , ,\
            &u_R & &textifquad f'(u_R), t leq x , ,
            endaligned
            right.
            tag2
            $$

            where the expression of the reciprocal $(f')^-1$ of $f'$ has been chosen carefully.


          • compound waves, a.k.a. composite waves or semi-shocks. The latter occur when neither shock waves nor rarefaction waves are entropy solutions, but combinations of them are. The position of rarefaction parts and of discontinuous parts is deduced from the Rankine-Hugoniot condition and from the Oleinik entropy condition.

          A rather practical method of solving such problems is convex hull construction: [1]




          The entropy-satisfying solution to a nonconvex Riemann problem can be determined from the graph of $f (u)$ in a simple manner. If $u_R < u_L$, then construct the convex hull of the set $lbrace (u, y) : u_R ≤ u ≤ u_L text and y ≤ f (u)rbrace$. The convex hull is the smallest convex set containing the original set. [...] If $u_L < u_R$, then the same idea works, but we look instead at the convex hull of the set of points above the graph, $lbrace (u, y) : u_L ≤ u ≤ u_R text and y ≥ f (u)rbrace$.




          Between $u_L$ and $u_R$, the intervals where the slope of the hull's edge is constant correspond to admissible discontinuities. The other intervals correspond to admissible rarefactions.



          One can also use Osher's expression of general similarity solutions $u(x,t) = v(xi)$, which writes [1]




          $$
          v(xi) =
          leftlbrace
          beginaligned
          &undersetu_Lleq uleq u_Rtextargmin left(f(u) - xi uright) && textifquad u_Lleq u_R , ,\
          &undersetu_Rleq uleq u_Ltextargmax left(f(u) - xi uright) && textifquad u_Rleq u_L , .
          endaligned
          right.
          $$





          To summarize, here are the different entropy solutions and their validity in the case $f(u) = frac13u^3$, where the inflection point of $f$ is located at the origin. The speed of sound is $f'(u) = u^2$, with reciprocal $(f')^-1(xi) = pmsqrtxi$. Using the convex hull construction method, one gets:



          • if $[0<u_L<u_R]$ or $[u_R<u_L<0]$, the solution is a rarefaction wave $(2)$ with shape $textsgn(u_R) sqrtx/t$.

          • else, if $[u_L<u_R< -frac12u_L]$ or $[-frac12u_L <u_R<u_L]$, the solution is a shock wave $(1)$, which speed $s = frac13left( u_L^2 + u_Lu_R + u_R^2 right)$ is given by the Rankine-Hugoniot condition.

          • else, if $[u_Lleq 0leq -frac12u_L leq u_R]$ or $[u_Rleq -frac12u_L leq 0 leq u_L]$, the solution is a semishock, more precisely a shock-rarefaction wave. The intermediate state $u^*$ which connects the discontinuous part to the rarefaction part satisfies $frac13left( u_L^2 + u_Lu^* + (u^*)^2 right) = (u^*)^2$ according to the convex hull construction, i.e. $u^* = -frac12u_L$. Thus,
            $$
            u(x,t) =
            leftlbrace
            beginaligned
            &u_L & &textifquad x leq left(-textstylefrac12u_Lright)^2, t , ,\
            &textsgn(u_R)sqrtx/t & &textifquad left(-textstylefrac12u_Lright)^2, t leq x leq u_R^2, t , ,\
            &u_R & &textifquad u_R^2, t leq x , .
            endaligned
            right.
            $$



          [1] R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, 2002.






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            5












            $begingroup$

            The method is very similar to the convex case, e.g. Burgers' equation where $f(u) = frac12u^2$, but there are more possible types of waves. In facts, in addition to shock waves and rarefaction waves, there may be waves with both discontinuous and continuous parts. Moreover, the Lax entropy condition for shocks must be replaced by the more general Oleinik entropy condition.



            In the case where the flux $f$ is not convex, these are the possible types of waves:




            • shock waves. If the solution is a shock wave with expression
              $$
              u(x,t) =
              leftlbrace
              beginaligned
              &u_L & &textifquad x < s, t , ,\
              &u_R & &textifquad s, t < x , ,
              endaligned
              right.
              tag1
              $$

              then the speed of shock $s$ must satisfy the Rankine-Hugoniot jump condition
              $s = fracf(u_R)- f(u_L)u_R - u_L$. Moreover, the shock wave must satisfy the Oleinik entropy condition [1]
              $$
              fracf(u)- f(u_L)u - u_L geq s geq fracf(u_R)- f(u)u_R - u ,
              $$

              for all $u$ between $u_L$ and $u_R$. In the case where $f$ is convex, the slope of its chords can be compared with its derivative using convexity inequalities. Thus, the classical Lax entropy condition $f'(u_L)>s>f'(u_R)$ is recovered, where $f'$ denotes the derivative of $f$.


            • rarefaction waves. The derivation is similar to the convex case, starting with the self-similarity Ansatz $u(x,t) = v(xi)$ where $xi = x/t$, which gives $f'(v(xi)) = xi$. In the nonconvex case, the equation $f'(v(xi)) = xi$ may have multiple solutions $v(xi)$, and the correct one is deduced from the continuity conditions $v(f'(u_L)) = u_L$ and $v(f'(u_R)) = u_R$. Such a solution is given by
              $$
              u(x,t) =
              leftlbrace
              beginaligned
              &u_L & &textifquad x leq f'(u_L), t , ,\
              &(f')^-1(x/t) & &textifquad f'(u_L), t leq x leq f'(u_R), t , ,\
              &u_R & &textifquad f'(u_R), t leq x , ,
              endaligned
              right.
              tag2
              $$

              where the expression of the reciprocal $(f')^-1$ of $f'$ has been chosen carefully.


            • compound waves, a.k.a. composite waves or semi-shocks. The latter occur when neither shock waves nor rarefaction waves are entropy solutions, but combinations of them are. The position of rarefaction parts and of discontinuous parts is deduced from the Rankine-Hugoniot condition and from the Oleinik entropy condition.

            A rather practical method of solving such problems is convex hull construction: [1]




            The entropy-satisfying solution to a nonconvex Riemann problem can be determined from the graph of $f (u)$ in a simple manner. If $u_R < u_L$, then construct the convex hull of the set $lbrace (u, y) : u_R ≤ u ≤ u_L text and y ≤ f (u)rbrace$. The convex hull is the smallest convex set containing the original set. [...] If $u_L < u_R$, then the same idea works, but we look instead at the convex hull of the set of points above the graph, $lbrace (u, y) : u_L ≤ u ≤ u_R text and y ≥ f (u)rbrace$.




            Between $u_L$ and $u_R$, the intervals where the slope of the hull's edge is constant correspond to admissible discontinuities. The other intervals correspond to admissible rarefactions.



            One can also use Osher's expression of general similarity solutions $u(x,t) = v(xi)$, which writes [1]




            $$
            v(xi) =
            leftlbrace
            beginaligned
            &undersetu_Lleq uleq u_Rtextargmin left(f(u) - xi uright) && textifquad u_Lleq u_R , ,\
            &undersetu_Rleq uleq u_Ltextargmax left(f(u) - xi uright) && textifquad u_Rleq u_L , .
            endaligned
            right.
            $$





            To summarize, here are the different entropy solutions and their validity in the case $f(u) = frac13u^3$, where the inflection point of $f$ is located at the origin. The speed of sound is $f'(u) = u^2$, with reciprocal $(f')^-1(xi) = pmsqrtxi$. Using the convex hull construction method, one gets:



            • if $[0<u_L<u_R]$ or $[u_R<u_L<0]$, the solution is a rarefaction wave $(2)$ with shape $textsgn(u_R) sqrtx/t$.

            • else, if $[u_L<u_R< -frac12u_L]$ or $[-frac12u_L <u_R<u_L]$, the solution is a shock wave $(1)$, which speed $s = frac13left( u_L^2 + u_Lu_R + u_R^2 right)$ is given by the Rankine-Hugoniot condition.

            • else, if $[u_Lleq 0leq -frac12u_L leq u_R]$ or $[u_Rleq -frac12u_L leq 0 leq u_L]$, the solution is a semishock, more precisely a shock-rarefaction wave. The intermediate state $u^*$ which connects the discontinuous part to the rarefaction part satisfies $frac13left( u_L^2 + u_Lu^* + (u^*)^2 right) = (u^*)^2$ according to the convex hull construction, i.e. $u^* = -frac12u_L$. Thus,
              $$
              u(x,t) =
              leftlbrace
              beginaligned
              &u_L & &textifquad x leq left(-textstylefrac12u_Lright)^2, t , ,\
              &textsgn(u_R)sqrtx/t & &textifquad left(-textstylefrac12u_Lright)^2, t leq x leq u_R^2, t , ,\
              &u_R & &textifquad u_R^2, t leq x , .
              endaligned
              right.
              $$



            [1] R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, 2002.






            share|cite|improve this answer











            $endgroup$

















              5












              $begingroup$

              The method is very similar to the convex case, e.g. Burgers' equation where $f(u) = frac12u^2$, but there are more possible types of waves. In facts, in addition to shock waves and rarefaction waves, there may be waves with both discontinuous and continuous parts. Moreover, the Lax entropy condition for shocks must be replaced by the more general Oleinik entropy condition.



              In the case where the flux $f$ is not convex, these are the possible types of waves:




              • shock waves. If the solution is a shock wave with expression
                $$
                u(x,t) =
                leftlbrace
                beginaligned
                &u_L & &textifquad x < s, t , ,\
                &u_R & &textifquad s, t < x , ,
                endaligned
                right.
                tag1
                $$

                then the speed of shock $s$ must satisfy the Rankine-Hugoniot jump condition
                $s = fracf(u_R)- f(u_L)u_R - u_L$. Moreover, the shock wave must satisfy the Oleinik entropy condition [1]
                $$
                fracf(u)- f(u_L)u - u_L geq s geq fracf(u_R)- f(u)u_R - u ,
                $$

                for all $u$ between $u_L$ and $u_R$. In the case where $f$ is convex, the slope of its chords can be compared with its derivative using convexity inequalities. Thus, the classical Lax entropy condition $f'(u_L)>s>f'(u_R)$ is recovered, where $f'$ denotes the derivative of $f$.


              • rarefaction waves. The derivation is similar to the convex case, starting with the self-similarity Ansatz $u(x,t) = v(xi)$ where $xi = x/t$, which gives $f'(v(xi)) = xi$. In the nonconvex case, the equation $f'(v(xi)) = xi$ may have multiple solutions $v(xi)$, and the correct one is deduced from the continuity conditions $v(f'(u_L)) = u_L$ and $v(f'(u_R)) = u_R$. Such a solution is given by
                $$
                u(x,t) =
                leftlbrace
                beginaligned
                &u_L & &textifquad x leq f'(u_L), t , ,\
                &(f')^-1(x/t) & &textifquad f'(u_L), t leq x leq f'(u_R), t , ,\
                &u_R & &textifquad f'(u_R), t leq x , ,
                endaligned
                right.
                tag2
                $$

                where the expression of the reciprocal $(f')^-1$ of $f'$ has been chosen carefully.


              • compound waves, a.k.a. composite waves or semi-shocks. The latter occur when neither shock waves nor rarefaction waves are entropy solutions, but combinations of them are. The position of rarefaction parts and of discontinuous parts is deduced from the Rankine-Hugoniot condition and from the Oleinik entropy condition.

              A rather practical method of solving such problems is convex hull construction: [1]




              The entropy-satisfying solution to a nonconvex Riemann problem can be determined from the graph of $f (u)$ in a simple manner. If $u_R < u_L$, then construct the convex hull of the set $lbrace (u, y) : u_R ≤ u ≤ u_L text and y ≤ f (u)rbrace$. The convex hull is the smallest convex set containing the original set. [...] If $u_L < u_R$, then the same idea works, but we look instead at the convex hull of the set of points above the graph, $lbrace (u, y) : u_L ≤ u ≤ u_R text and y ≥ f (u)rbrace$.




              Between $u_L$ and $u_R$, the intervals where the slope of the hull's edge is constant correspond to admissible discontinuities. The other intervals correspond to admissible rarefactions.



              One can also use Osher's expression of general similarity solutions $u(x,t) = v(xi)$, which writes [1]




              $$
              v(xi) =
              leftlbrace
              beginaligned
              &undersetu_Lleq uleq u_Rtextargmin left(f(u) - xi uright) && textifquad u_Lleq u_R , ,\
              &undersetu_Rleq uleq u_Ltextargmax left(f(u) - xi uright) && textifquad u_Rleq u_L , .
              endaligned
              right.
              $$





              To summarize, here are the different entropy solutions and their validity in the case $f(u) = frac13u^3$, where the inflection point of $f$ is located at the origin. The speed of sound is $f'(u) = u^2$, with reciprocal $(f')^-1(xi) = pmsqrtxi$. Using the convex hull construction method, one gets:



              • if $[0<u_L<u_R]$ or $[u_R<u_L<0]$, the solution is a rarefaction wave $(2)$ with shape $textsgn(u_R) sqrtx/t$.

              • else, if $[u_L<u_R< -frac12u_L]$ or $[-frac12u_L <u_R<u_L]$, the solution is a shock wave $(1)$, which speed $s = frac13left( u_L^2 + u_Lu_R + u_R^2 right)$ is given by the Rankine-Hugoniot condition.

              • else, if $[u_Lleq 0leq -frac12u_L leq u_R]$ or $[u_Rleq -frac12u_L leq 0 leq u_L]$, the solution is a semishock, more precisely a shock-rarefaction wave. The intermediate state $u^*$ which connects the discontinuous part to the rarefaction part satisfies $frac13left( u_L^2 + u_Lu^* + (u^*)^2 right) = (u^*)^2$ according to the convex hull construction, i.e. $u^* = -frac12u_L$. Thus,
                $$
                u(x,t) =
                leftlbrace
                beginaligned
                &u_L & &textifquad x leq left(-textstylefrac12u_Lright)^2, t , ,\
                &textsgn(u_R)sqrtx/t & &textifquad left(-textstylefrac12u_Lright)^2, t leq x leq u_R^2, t , ,\
                &u_R & &textifquad u_R^2, t leq x , .
                endaligned
                right.
                $$



              [1] R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, 2002.






              share|cite|improve this answer











              $endgroup$















                5












                5








                5





                $begingroup$

                The method is very similar to the convex case, e.g. Burgers' equation where $f(u) = frac12u^2$, but there are more possible types of waves. In facts, in addition to shock waves and rarefaction waves, there may be waves with both discontinuous and continuous parts. Moreover, the Lax entropy condition for shocks must be replaced by the more general Oleinik entropy condition.



                In the case where the flux $f$ is not convex, these are the possible types of waves:




                • shock waves. If the solution is a shock wave with expression
                  $$
                  u(x,t) =
                  leftlbrace
                  beginaligned
                  &u_L & &textifquad x < s, t , ,\
                  &u_R & &textifquad s, t < x , ,
                  endaligned
                  right.
                  tag1
                  $$

                  then the speed of shock $s$ must satisfy the Rankine-Hugoniot jump condition
                  $s = fracf(u_R)- f(u_L)u_R - u_L$. Moreover, the shock wave must satisfy the Oleinik entropy condition [1]
                  $$
                  fracf(u)- f(u_L)u - u_L geq s geq fracf(u_R)- f(u)u_R - u ,
                  $$

                  for all $u$ between $u_L$ and $u_R$. In the case where $f$ is convex, the slope of its chords can be compared with its derivative using convexity inequalities. Thus, the classical Lax entropy condition $f'(u_L)>s>f'(u_R)$ is recovered, where $f'$ denotes the derivative of $f$.


                • rarefaction waves. The derivation is similar to the convex case, starting with the self-similarity Ansatz $u(x,t) = v(xi)$ where $xi = x/t$, which gives $f'(v(xi)) = xi$. In the nonconvex case, the equation $f'(v(xi)) = xi$ may have multiple solutions $v(xi)$, and the correct one is deduced from the continuity conditions $v(f'(u_L)) = u_L$ and $v(f'(u_R)) = u_R$. Such a solution is given by
                  $$
                  u(x,t) =
                  leftlbrace
                  beginaligned
                  &u_L & &textifquad x leq f'(u_L), t , ,\
                  &(f')^-1(x/t) & &textifquad f'(u_L), t leq x leq f'(u_R), t , ,\
                  &u_R & &textifquad f'(u_R), t leq x , ,
                  endaligned
                  right.
                  tag2
                  $$

                  where the expression of the reciprocal $(f')^-1$ of $f'$ has been chosen carefully.


                • compound waves, a.k.a. composite waves or semi-shocks. The latter occur when neither shock waves nor rarefaction waves are entropy solutions, but combinations of them are. The position of rarefaction parts and of discontinuous parts is deduced from the Rankine-Hugoniot condition and from the Oleinik entropy condition.

                A rather practical method of solving such problems is convex hull construction: [1]




                The entropy-satisfying solution to a nonconvex Riemann problem can be determined from the graph of $f (u)$ in a simple manner. If $u_R < u_L$, then construct the convex hull of the set $lbrace (u, y) : u_R ≤ u ≤ u_L text and y ≤ f (u)rbrace$. The convex hull is the smallest convex set containing the original set. [...] If $u_L < u_R$, then the same idea works, but we look instead at the convex hull of the set of points above the graph, $lbrace (u, y) : u_L ≤ u ≤ u_R text and y ≥ f (u)rbrace$.




                Between $u_L$ and $u_R$, the intervals where the slope of the hull's edge is constant correspond to admissible discontinuities. The other intervals correspond to admissible rarefactions.



                One can also use Osher's expression of general similarity solutions $u(x,t) = v(xi)$, which writes [1]




                $$
                v(xi) =
                leftlbrace
                beginaligned
                &undersetu_Lleq uleq u_Rtextargmin left(f(u) - xi uright) && textifquad u_Lleq u_R , ,\
                &undersetu_Rleq uleq u_Ltextargmax left(f(u) - xi uright) && textifquad u_Rleq u_L , .
                endaligned
                right.
                $$





                To summarize, here are the different entropy solutions and their validity in the case $f(u) = frac13u^3$, where the inflection point of $f$ is located at the origin. The speed of sound is $f'(u) = u^2$, with reciprocal $(f')^-1(xi) = pmsqrtxi$. Using the convex hull construction method, one gets:



                • if $[0<u_L<u_R]$ or $[u_R<u_L<0]$, the solution is a rarefaction wave $(2)$ with shape $textsgn(u_R) sqrtx/t$.

                • else, if $[u_L<u_R< -frac12u_L]$ or $[-frac12u_L <u_R<u_L]$, the solution is a shock wave $(1)$, which speed $s = frac13left( u_L^2 + u_Lu_R + u_R^2 right)$ is given by the Rankine-Hugoniot condition.

                • else, if $[u_Lleq 0leq -frac12u_L leq u_R]$ or $[u_Rleq -frac12u_L leq 0 leq u_L]$, the solution is a semishock, more precisely a shock-rarefaction wave. The intermediate state $u^*$ which connects the discontinuous part to the rarefaction part satisfies $frac13left( u_L^2 + u_Lu^* + (u^*)^2 right) = (u^*)^2$ according to the convex hull construction, i.e. $u^* = -frac12u_L$. Thus,
                  $$
                  u(x,t) =
                  leftlbrace
                  beginaligned
                  &u_L & &textifquad x leq left(-textstylefrac12u_Lright)^2, t , ,\
                  &textsgn(u_R)sqrtx/t & &textifquad left(-textstylefrac12u_Lright)^2, t leq x leq u_R^2, t , ,\
                  &u_R & &textifquad u_R^2, t leq x , .
                  endaligned
                  right.
                  $$



                [1] R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, 2002.






                share|cite|improve this answer











                $endgroup$



                The method is very similar to the convex case, e.g. Burgers' equation where $f(u) = frac12u^2$, but there are more possible types of waves. In facts, in addition to shock waves and rarefaction waves, there may be waves with both discontinuous and continuous parts. Moreover, the Lax entropy condition for shocks must be replaced by the more general Oleinik entropy condition.



                In the case where the flux $f$ is not convex, these are the possible types of waves:




                • shock waves. If the solution is a shock wave with expression
                  $$
                  u(x,t) =
                  leftlbrace
                  beginaligned
                  &u_L & &textifquad x < s, t , ,\
                  &u_R & &textifquad s, t < x , ,
                  endaligned
                  right.
                  tag1
                  $$

                  then the speed of shock $s$ must satisfy the Rankine-Hugoniot jump condition
                  $s = fracf(u_R)- f(u_L)u_R - u_L$. Moreover, the shock wave must satisfy the Oleinik entropy condition [1]
                  $$
                  fracf(u)- f(u_L)u - u_L geq s geq fracf(u_R)- f(u)u_R - u ,
                  $$

                  for all $u$ between $u_L$ and $u_R$. In the case where $f$ is convex, the slope of its chords can be compared with its derivative using convexity inequalities. Thus, the classical Lax entropy condition $f'(u_L)>s>f'(u_R)$ is recovered, where $f'$ denotes the derivative of $f$.


                • rarefaction waves. The derivation is similar to the convex case, starting with the self-similarity Ansatz $u(x,t) = v(xi)$ where $xi = x/t$, which gives $f'(v(xi)) = xi$. In the nonconvex case, the equation $f'(v(xi)) = xi$ may have multiple solutions $v(xi)$, and the correct one is deduced from the continuity conditions $v(f'(u_L)) = u_L$ and $v(f'(u_R)) = u_R$. Such a solution is given by
                  $$
                  u(x,t) =
                  leftlbrace
                  beginaligned
                  &u_L & &textifquad x leq f'(u_L), t , ,\
                  &(f')^-1(x/t) & &textifquad f'(u_L), t leq x leq f'(u_R), t , ,\
                  &u_R & &textifquad f'(u_R), t leq x , ,
                  endaligned
                  right.
                  tag2
                  $$

                  where the expression of the reciprocal $(f')^-1$ of $f'$ has been chosen carefully.


                • compound waves, a.k.a. composite waves or semi-shocks. The latter occur when neither shock waves nor rarefaction waves are entropy solutions, but combinations of them are. The position of rarefaction parts and of discontinuous parts is deduced from the Rankine-Hugoniot condition and from the Oleinik entropy condition.

                A rather practical method of solving such problems is convex hull construction: [1]




                The entropy-satisfying solution to a nonconvex Riemann problem can be determined from the graph of $f (u)$ in a simple manner. If $u_R < u_L$, then construct the convex hull of the set $lbrace (u, y) : u_R ≤ u ≤ u_L text and y ≤ f (u)rbrace$. The convex hull is the smallest convex set containing the original set. [...] If $u_L < u_R$, then the same idea works, but we look instead at the convex hull of the set of points above the graph, $lbrace (u, y) : u_L ≤ u ≤ u_R text and y ≥ f (u)rbrace$.




                Between $u_L$ and $u_R$, the intervals where the slope of the hull's edge is constant correspond to admissible discontinuities. The other intervals correspond to admissible rarefactions.



                One can also use Osher's expression of general similarity solutions $u(x,t) = v(xi)$, which writes [1]




                $$
                v(xi) =
                leftlbrace
                beginaligned
                &undersetu_Lleq uleq u_Rtextargmin left(f(u) - xi uright) && textifquad u_Lleq u_R , ,\
                &undersetu_Rleq uleq u_Ltextargmax left(f(u) - xi uright) && textifquad u_Rleq u_L , .
                endaligned
                right.
                $$





                To summarize, here are the different entropy solutions and their validity in the case $f(u) = frac13u^3$, where the inflection point of $f$ is located at the origin. The speed of sound is $f'(u) = u^2$, with reciprocal $(f')^-1(xi) = pmsqrtxi$. Using the convex hull construction method, one gets:



                • if $[0<u_L<u_R]$ or $[u_R<u_L<0]$, the solution is a rarefaction wave $(2)$ with shape $textsgn(u_R) sqrtx/t$.

                • else, if $[u_L<u_R< -frac12u_L]$ or $[-frac12u_L <u_R<u_L]$, the solution is a shock wave $(1)$, which speed $s = frac13left( u_L^2 + u_Lu_R + u_R^2 right)$ is given by the Rankine-Hugoniot condition.

                • else, if $[u_Lleq 0leq -frac12u_L leq u_R]$ or $[u_Rleq -frac12u_L leq 0 leq u_L]$, the solution is a semishock, more precisely a shock-rarefaction wave. The intermediate state $u^*$ which connects the discontinuous part to the rarefaction part satisfies $frac13left( u_L^2 + u_Lu^* + (u^*)^2 right) = (u^*)^2$ according to the convex hull construction, i.e. $u^* = -frac12u_L$. Thus,
                  $$
                  u(x,t) =
                  leftlbrace
                  beginaligned
                  &u_L & &textifquad x leq left(-textstylefrac12u_Lright)^2, t , ,\
                  &textsgn(u_R)sqrtx/t & &textifquad left(-textstylefrac12u_Lright)^2, t leq x leq u_R^2, t , ,\
                  &u_R & &textifquad u_R^2, t leq x , .
                  endaligned
                  right.
                  $$



                [1] R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, 2002.







                share|cite|improve this answer














                share|cite|improve this answer



                share|cite|improve this answer








                edited Mar 25 at 7:00

























                answered Nov 27 '17 at 15:13









                Harry49Harry49

                8,76331346




                8,76331346



























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